Combinatorial and geometric group theory: Dortmund and Ottawa-Montreal conf

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Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference. Proposals for volumes can be sent to the Mathematics Editor at either Springer Basel AG Birkhäuser P.O. Box 133 CH-4010 Basel Switzerland or Birkhauser Boston 233 Spring Street New York, NY 10013 USA

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Combinatorial and Geometric Group Theory Dortmund and Ottawa-Montreal Conferences

Oleg Bogopolski Inna Bumagin Olga Kharlampovich Enric Ventura Editors

Birkhäuser

Editors: Oleg Bogopolski Mathematisches Institut der Heinrich-Heine-Universität Düsseldorf Universitätsstr. 1 40225 Düsseldorf Germany e-mail: [email protected]

Olga Kharlampovich Department of Mathematics and Statistics McGill University 805 Sherbrooke St., West Montreal, Quebec, H3A 2K6 Canada e-mail: [email protected]

Inna Bumagin School of Mathematics and Statistics Carleton University 1125 Colonel By Drive Ottawa, Ontario, K1S 5B6 Canada e-mail: [email protected]

Enric Ventura Departament de Matematica Aplicada III EPSEM – Universitat Politècnica de Catalunya Av. Bases de Manresa 61–73 08242 Manresa, Barcelona Spain e-mail: [email protected]

2000 Mathematics Subject Classiﬁcation 20A, 20E, 20F, 20H, 20M, 20P, 03B, 03D, 05C, 08A, 51F, 57M, 57S, 60B, 68Q Library of Congress Control Number: 2010926413 Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliograﬁe; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

ISBN 978-3-7643-9910-8 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microﬁlms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2010 Springer Basel AG P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Cover Design: Alexander Faust, Basel, Switzerland Printed in Germany ISBN 978-3-7643-9910-8

e-ISBN 978-3-7643-9911-5

987654321

www.birkhauser.ch

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

O. Bogopolski and R. Vikentiev Subgroups of Small Index in Aut(Fn ) and Kazhdan’s Property (T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

P. Brinkmann Dynamics of Free Group Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

V. Diekert, A.J. Duncan and A.G. Myasnikov Geodesic Rewriting Systems and Pregroups . . . . . . . . . . . . . . . . . . . . . . . . . .

55

E. Frenkel, A.G. Myasnikov and V.N. Remeslennikov Regular Sets and Counting in Free Groups . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

D. Gon¸calves and P. Wong Twisted Conjugacy for Virtually Cyclic Groups and Crystallographic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 M. Hock and B. Tsaban Solving Random Equations in Garside Groups Using Length Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 A. Juh´ asz An Application of Word Combinatorics to Decision Problems in Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171

O. Kharlampovich and A.G. Myasnikov Equations and Fully Residually Free Groups . . . . . . . . . . . . . . . . . . . . . . . . . 203 M. Lustig The FN -action on the Product of the Two Limit Trees for an Iwip Automorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 F. Matucci Mather Invariants in Groups of Piecewise-linear Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

vi

Contents

P.V. Morar and A.N. Shevlyakov Algebraic Geometry over the Additive Monoid of Natural Numbers: Systems of Coeﬃcient Free Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 D. Savchuk Some Graphs Related to Thompson’s Group F . . . . . . . . . . . . . . . . . . . . . .

279

R. Weidmann Generating Tuples of Virtually Free Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 297 R. Zarzycki Limits of Thompson’s Group F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

Combinatorial and Geometric Group Theory Trends in Mathematics, vii–viii c 2010 Springer Basel AG

Preface We are pleased to present the book “Geometric Group Theory, Dortmund and Carleton Conferences”, a selection of the best research articles from two strongly related 2007 international conferences: • “Combinatorial and Geometric Group Theory with Applications” (GAGTA), the University of Dortmund (Germany) from August 27th to 31st; • “Fields Workshop in Asymptotic Group Theory and Cryptography”, Carleton University (Ottawa, Canada) from December 14th to 16th, followed by “Workshop on Actions on Trees, Non-Archimedian Words, and Asymptotic Cones”, Saint Sauveur (Montreal) from December 17th to 21st. The book contains a selection of refereed papers on Combinatorial and Geometric Group Theory. The breadth of topics included will assure the interest of all specialists and researchers in this area of mathematics; they will also prove to be valuable to graduate students and mathematicians in other areas who wish to explore deeper into this exciting and very active ﬁeld of research. The articles largely fall into ﬁve categories: • equations and algebraic geometry over groups; Tarski problems, • algorithmic problems in groups, • groups of automorphisms of non-abelian free groups, • groups of transformations of the unit interval and Thompson’s group F , • questions motivated by group-based cryptography. Readers interested in the ﬁrst topic may choose to look ﬁrst at the excellent expository paper by O. Kharlampovich and A.G. Myasnikov. Here, the authors explain their multifaceted techniques (part of them on algebraic geometry over groups) for solving two of Tarski’s famous problems on elementary theories of free groups. The paper of P. Morar and A. Shevlyakov initiates investigations of algebraic geometry over some intriquing classes of monoids. One can also learn a lot about dynamics of automorphisms of free groups via train tracks and actions on trees, by reading the thought-provoking papers of P. Brinkmann and M. Lustig. In a similar direction, R. Weidmann shows how Makanin-Razborov diagrams and Stallings foldings can be used to solve the rank problem for virtually free groups.

viii

Preface

The paper by O. Bogopolski and A. Vikentiev describes some particularly useful ﬁnite index subgroups of the automorphism group of a ﬁnitely generated free group. One of their uses may be to attack the problem on the Kazhdan property (T ) for these groups. The paper of A. Juhasz contains a solution of the diﬃcult membership problem in a subclass of one-relator groups. Papers of F. Matucci, D. Savchuk and R. Zarzycki will attract the attention of those who want to know more about groups of transformations of the unit interval [0, 1], in particular about the famous Thompson’s group F and its limit properties. The paper by A.J. Duncan, V. Dieckert and A.G. Myasnikov contains a very thorough survey on rewriting systems with new issues on inﬁnite rewriting systems. The paper by L. Frenkel, A.G. Myasnikov and V.N. Remeslennikov is devoted to the problem of how to measure some subsets in free groups by using random walks. The results of this paper may be used for designing algorithms that run fast on almost all inputs. This paper as well as the paper by M. Hock and B. Tsaban are highly recommended to specialists in cryptography. Finally, the paper by D. Goncalves and P. Wong is devoted to the twisted conjugacy in 2-dimensional crystallographic groups. We are very grateful to the organizations that supported these two conferences: • The conference in Dortmund was organized by O. Bogopolski, M.-T. Bochnig, G. Rosenberger, V. Shpilrain and E. Ventura. This conference was ﬁnancially supported by DAAD (Deutscher Akademischer Austauschdienst), by DFG (Deutsche Forschungsgemeinschaft), and by the Universit¨ at Dortmund. The URL address for its homepage is http://www.mathematik.uni-dortmund.de/∼gcgta/. • The workshops in Canada were co-organized by I. Bumagin, O. Kharlampovich and A.G. Myasnikov. The workshops could not have been held without the generous support of the Fields Institute. The organizers also gratefully acknowledge the ﬁnancial support provided by the Faculty of Science of Carleton University and by McGill University. More information about the workshops can be found at the URL http://www.fields.utoronto.ca/programs/ scientific/07-08/asympotic/index.html Finally, we wish to thank the contributors to this volume, and the anonymous referees who ensured the high quality of its contents. Our thanks also go to Thomas Hempﬂing at Birkh¨ auser for his assistance in the typsetting and preparation of this volume. Without these joint eﬀorts, this book would never have appeared. The editors,

O. I. O. E.

Bogopolski, Bumagin, Kharlampovich, Ventura

Combinatorial and Geometric Group Theory Trends in Mathematics, 1–17 c 2010 Springer Basel AG

Subgroups of Small Index in Aut(Fn) and Kazhdan’s Property (T) O. Bogopolski and R. Vikentiev Abstract. We introduce a series of interesting subgroups of ﬁnite index in Aut(Fn ). One of them has index 42 in Aut(F3 ) and inﬁnite abelianization. This implies that Aut(F3 ) does not have Kazhdan’s property (T); see [15] and [5] for other proofs. We prove also that every subgroup of ﬁnite index in Aut(Fn ), n 3, which contains the subgroup of IA-automorphisms has a ﬁnite abelianization. We introduce a subgroup K(n) of ﬁnite index in Aut(Fn ) and show, that its abelianization is inﬁnite for n = 3, and it is ﬁnite for n 4. We ask, whether the abelianization of its commutator subgroup K(n) is inﬁnite for n 4. If so, then Aut(Fn ) would not have Kazhdan’s property (T) for n 4. Mathematics Subject Classiﬁcation (2000). 20F28, 20E05, 20E15. Keywords. Automorphisms, free groups, Kazhdan’s property (T), congruence subgroups.

1. Deﬁnitions, problems and motivations In the mid 60’s, D. Kazhdan deﬁned his property (T) for locally compact groups and used it as a tool to demonstrate that many lattices in these groups are ﬁnitely generated [8]. Later this property found various surprising applications, in particular, in the ﬁrst explicit construction of expander graphs by G. Margulis [14], see also the book of A. Lubotzky [9] and the paper of A. Lubotzky and I. Pak [10]. We recommend to the reader the very informative book of B. Bekka, P. de la Harpe and A. Valette [2] and the lecture of Y. Shalom [21] on the property (T) and its applications. There are several equivalent deﬁnitions of the property (T) for topological groups. Below we give one of them in case, where the group is ﬁnitely generated. We will assume that the group is endowed with the discrete topology. Let G be a ﬁnitely generated group and π : G → U(H) a unitary representation of G on a Hilbert space H. If S is a ﬁnite generating set of G and > 0, then

2

O. Bogopolski and R. Vikentiev

a unit vector v in H is called an (S, )-invariant vector if ||π(s)(v) − v|| < for all s ∈ S. Deﬁnition 1.1. A ﬁnitely generated group G has the property (T) iﬀ for every ﬁnite generating subset S ⊆ G there exists some > 0, such that the following is satisﬁed: For any unitary representation π : G → U(H) on a Hilbert space H, the existence of an (S, )-invariant vector implies the existence of a non-zero π(G)-invariant vector. Any such is called a Kazhdan constant for G with respect to S. It is easy to show, that in this deﬁnition the words “for every ﬁnite generating set” can be replaced by “for some ﬁnite generating set”. Deﬁnition 1.2. A ﬁnitely generated group G has property (FH) if any action of G by aﬃne isometries on a Hilbert space has a ﬁxed point. By a theorem of Delorme–Guichardet (see Theorem 2.12.4 in [2]), the properties (T) and (FH) are equivalent for ﬁnitely generated groups. There are strong consequences on several types of actions: for a group with the property (FH), any isometric action on a tree has a ﬁxed point or a ﬁxed edge (this is the property (FA) of Serre), any isometric action on a real or complex hyperbolic space has a ﬁxed point. Deﬁnition 1.3. A group G has Serre’s property (FA) if acting simplicially and without inversions of edges on any simplicial tree, G has a global ﬁxed point. If G is ﬁnitely generated, this property can be reformulated in purely algebraic terms. Theorem 1.4. (J.-P. Serre; 1974). A ﬁnitely generated group G has the property (FA) if and only if the following two statements hold: (1) G is not a nontrivial amalgamated product, that is G A ∗C B with C = A and C = B. (2) G does not have a quotient isomorphic to Z. Theorem 1.5. (Y. Watatani; 1982). Let G be a countable group. If G has the property (T) of Kazhdan, then it has the property (FA) of Serre. Let Fn be the free group of rank n with basis x1 , . . . , xn . In this paper we will concentrate on the group Aut(Fn ), the automorphism group of Fn . There is the canonical epimorphism Φ : Aut(Fn ) → GLn (Z), which sends an automorphism α ∈ Aut(Fn ) to the matrix, whose ij-entry equals to the sum of exponents of xj in α(xi ), i, j = 1, . . . , n. The full preimage of SLn (Z) is called the special automorphism group of Fn and is denoted by SAut(Fn ). The kernel of Φ is denoted by IA(Fn ).

Subgroups of Small Index in Aut(Fn )

3

In the following table we summarize known facts on the (T) and (FA) properties for SLn (Z) and SAut(Fn ), n 3. (T)

⇒ (FA)

SLn (Z), n 3

+ (Kazhdan, 1967)

+ (Serre, 1974)

SAut (F3 ),

− (McCool, 1989)

+ (Bogopolski, 1987)

SAut (Fn ), n 4,

?

+ (Bogopolski, 1987)

Remark 1.6. The property (T) is preserved under taking subgroups of ﬁnite index, the property (FA) is not preserved. Both properties are preserved under taking overgroups of ﬁnite index. Groups with the property (T) have no subgroups of ﬁnite index, which can be mapped onto Z. Problems 1.7. 1) Does the group Aut(Fn ), n 4, have the property (T)? 2) Does every ﬁnite index subgroup of Aut(Fn ), n 4, have the property (FA)? 3) Characterize (in terms of actions on trees or algebraically) those ﬁnitely generated groups, whose subgroups of ﬁnite index have the (FA) property. In [23], K. Vogtmann formulated the Out-versions of the problems 1) and 2) (see Problems 14 and 15 there). The ﬁrst problem is also formulated as Problem (7.1) in the list of open questions in [2]. Due to A. Lubotzky and I. Pak [10], the presence of the Kazhdan property for Aut(Fn ) would imply a very elegant construction of an inﬁnite series of εexpanders. The later have applications to theoretical computer science, design of robust computer networks, and the theory of error-correcting codes [7]. Deﬁnition 1.8. Let ε be a positive real number. A ﬁnite d-regular graph Γ is called an ε-expander, if for every subset of vertices B ⊂ Γ0 with |B| |Γ0 |/2, we have |∂B| ε|B|, where ∂B = {v ∈ Γ0 | v ∈ / B, but v is adjacent to a vertex in B}. Deﬁnition 1.9 (Graph Γn (G)). Fix a natural number n and a ﬁnite group G, which can be generated by n elements. The vertices of the graph Γn (G) are all tuples (g) = (g1 , . . . , gn ) such that g1 , . . . , gn = G. Two tuples (g) and (g ) are connected by an edge if (g ) can be obtained from (g) by applying one of the following replacement operations: ± : (g1 , . . . , gi , . . . , gn ) → (g1 , . . . , gi · gj±1 , . . . , gn ), Ri,j ±1 L± i,j : (g1 , . . . , gi , . . . , gn ) → (g1 , . . . , gj · gi , . . . , gn ),

where i = j.

4

O. Bogopolski and R. Vikentiev

Clearly, the graph Γn (G) is 4n(n − 1)-regular. The following theorem is a partial case of a more general Theorem 3.1 in [10]. It explains why establishing the Kazhdan property (T) for Aut(Fn ) is important. Theorem 1.10 (A. Lubotzky and I. Pak; 2001). If the group Aut(Fn ) (or equivalently SAut(Fn )) has the property (T), then there exists an ε > 0, such that every connected component of the graph Γn (G) is an ε-expander for any n-generated ﬁnite group G. The group Aut(F2 ) does not have the property (T), since it can be represented as a nontrivial amalgamated product. (The uniqueness of such representation up to conjugacy was shown by O. Bogopolski in [3]). The following two theorems imply that Aut(F3 ) does not have the property (T). In this paper we suggest some ideas for studying this problem in rank n 4. Theorem 1.11 (J. McCool; 1989). There is a subgroup of ﬁnite index in Out(F3 ), which can be approximated by torsion-free nilpotent groups. In particular, this subgroup can be mapped onto Z. Therefore Out(F3 ) and Aut(F3 ) do not have the Kazhdan property (T). Theorem 1.12 (F. Grunewald and A. Lubotzky; 2006). There exists a subgroup of index 168 in Aut(F3 ) which can be mapped onto F2 . In particular, Aut(F3 ) has no Kazhdan’s property (T). The proof of this theorem has existed for a long time in a folklore form. We know variants of the proof from private talks with A. Casson (2000) and M. Bridson (2004). To our best knowledge, the ﬁrst written proof (based on the same idea) appeared in the paper of F. Grunewald and A. Lubotzky [5], see Corollary 1.3 there. This corollary is deduced there from a more general Theorem 1.1 on representations of Aut(Fn ). The structure of the paper is as follows. In Section 2 we give a short exposition of the proof of Theorem 1.12. Some elements of this proof motivated us to further constructions. In Section 3 we introduce some useful automorphisms. In Section 4 we prove that any ﬁnite index subgroup of Aut(Fn ) containing IA(Fn ) has ﬁnite abelianization (Theorem 4.1). Thus, to construct a ﬁnite index subgroup in Aut(Fn ) with inﬁnite abelianization, one should avoid IA-automorphisms. In Section 5 we introduce and investigate congruence subgroups Cong(n, m) and SCong(n, m) in Aut(Fn ) and SAut(Fn ) respectively. Both congruence subgroups contain IA(Fn ). In Section 6 we deﬁne a subgroup K(n) of index 2 in SCong(n, 2), so that it does not contain IA(Fn ). In Section 7 we show that K(3) has inﬁnite abelianization. This gives an alternative proof of Theorem 1.12. Further we construct a series of overgroups of K(3): Aut(F3 ) C(3) B(3) A(3) K(3). 42

2

2

2

The largest one, C(3), has index 42 in Aut(F3 ) and inﬁnite abelianization. We conjecture, that this is the minimal possible index for a subgroup in Aut(F3 )

Subgroups of Small Index in Aut(Fn )

5

with inﬁnite abelianization. We compute the abelianizations of these subgroups explicitly: K(3)/K(3) ∼ = Z14 × Z × Z, 2

A(3)/A(3) ∼ = Z72 × Z × Z, B(3)/B(3) ∼ = Z4 × Z, 2

C(3)/C(3) ∼ = Z32 × Z4 × Z.

Here we use the notation Zkn = Z/nZ × · · · × Z/nZ. k times

In Section 8 we prove that if n 4, then the abelianization of K(n) is a ﬁnite 2-group. In particular, the abelianization of K(4) is Z38 2 . We would like to know, what is the abelianization of the commutator subgroup K(n) for n 4. If it is inﬁnite, the group Aut(Fn ) does not have Kazhdan’s property (T) for n 4. In this paper we use the following notation for the commutator: [x, y] = xyx−1 y −1 .

2. A sketch of the proof of F. Grunewald and A. Lubotzky that Aut(F3) has no Kazhdan’s property (T) Let F3 = F (a, b, c) be the free group on free generators a, b, c. There exist exactly 7 epimorphisms F (a, b, c) → Z2 . Therefore there exist exactly 7 subgroups of index (1) (7) 2 in F3 . Denote them by F5 , . . . , F5 . Clearly, every such subgroup has rank 5. (1) We will work with one of them, F5 = a, b, c2 , c−1 ac, c−1 bc. It is easy to check that Aut(F3 ) acts transitively on the set of these subgroups. Therefore the index (1) of St(F5 ) in Aut(F3 ) is 7, where (1)

(1)

(1)

St(F5 ) = α ∈ Aut(F3 ) | α(F5 ) = F5 . (1)

(1)

Clearly, the restriction map Res : St(F5 ) → Aut(F5 ), α → α|F (1) , is an 5

embedding. Now we introduce an important inner automorphism τ : x → c−1 xc, x ∈ F3 . Lemma 2.1. (1)

1) τ ∈ St(F5 ), (1) 2) τ |F (1) ∈ / Inn(F5 ), 5

3) (τ −1 ϕ−1 τ ϕ)|F (1) ∈ Inn(F5 ) for every ϕ ∈ St(F5 ). (1)

(1)

5

Proof. The ﬁrst two statements are straightforward. We prove the third one. Let (1) (1) x ∈ F5 and ϕ ∈ St(F5 ). An easy computation shows that xτ −1 ϕ−1 τ ϕ = (1) ϕ(c−1 )cxc−1 ϕ(c). Thus we need to show that c−1 ϕ(c) ∈ F5 . The last is evident, (1) (1) since F5 and hence the second coset cF5 are ϕ-invariant.

6

O. Bogopolski and R. Vikentiev Consider the composition of two homomorphisms Res (1) (1) Ψ : St(F5 ) −→ Aut(F5 ) −→ GL5 (Z), (1)

where the second homomorphism sends an automorphism of F5 to the automor(1) phism induced on the abelianization of F5 (we identify the last automorphism (1) with the corresponding matrix, using the prescribed basis of F5 ). One can easily compute, that ⎞ ⎛ 0 0 0 1 0 ⎜ 0 0 0 0 1 ⎟ ⎟ ⎜ ⎟ Ψ(τ ) = ⎜ ⎜ 0 0 1 0 0 ⎟. ⎝ 1 0 0 0 0 ⎠ 0 1 0 0 0 Since (Ψ(τ ))2 = Id, we have Z5 ⊃ V+ ⊕ V− , where V+ = Ker(Ψ(τ ) + Id),

V− = Ker(Ψ(τ ) − Id).

The Z-submodule V+ has the basis {(1, 0, 0, −1, 0), (0, 1, 0, 0, −1)}. (1) By Lemma 2.1.3), the matrices Ψ(τ ) and Ψ(ϕ) commute for all ϕ ∈ St(F5 ). (1) Hence the submodule V+ is Ψ(ϕ)-invariant for every ϕ ∈ St(F5 ). Thus, there is the natural homomorphism (1) θ : St(F5 ) → GL(V+ ) ∼ = GL2 (Z), θ(ϕ) = Ψ(ϕ) . V+

This homomorphism is onto, since the automorphisms ϕ1 : a → a, b → ba, c → c (1) and ϕ2 : a → b, b → a, c → c belong to St(F5 ) and are mapped onto the matrices

0 1 1 0 , , 1 0 1 1 which generate GL2 (Z) Notice that GL2 (Z) ∼ = D4 ∗D2 D6 , where Dm denotes the dihedral group of order 2m (see [4] for example). Therefore there exists an epimorphism μ : GL2 (Z) → D12 . The kernel of μ is a free group of rank 2, we denote it by F2 . Thus we have the following chain of embeddings and epimorphisms: (1)

θ

μ

Aut(F3 ) St(F5 ) → GL2 (Z) → D12 . (1)

Let H = Ker(θμ). Then H has index 24 in St(F5 ). Hence H has index 168 in Aut(F3 ). Moreover, θ(H) = Ker(μ) = F2 . In particular, H can be homomorphically mapped onto Z. Hence Aut(F3 ) does not have the Kazhdan property (T). Remark 2.2. The group H is not normal in Aut(F3 ). The above construction cannot be generalized for Aut(Fn ), where n 4, since in this case GLn−1 (Z) (contrary to GL2 (Z)) does not contain a subgroup of ﬁnite index with inﬁnite abelianization.

Subgroups of Small Index in Aut(Fn )

7

3. Some notations and useful automorphisms Let Fn be the free group on free generators x1 , x2 , . . . , xn . First we deﬁne some automorphisms of Fn . We will write the image of xi only if it diﬀers from xi . 1) For any i, j, k ∈ {1, 2, . . . , n}, where k = i, j, we deﬁne the automorphism −1 αijk : xi → xi · x−1 j xk xj xk .

In particular, αiik : xi → x−1 k xi xk . Note that αijk = α−1 ikj for distinct i, j, k. We say that the automorphism αijk is of the ﬁrst kind if i, j, k are distinct, and of the second kind if i = j. 2) For any i, j ∈ {1, 2, . . . , n}, where i = j, we deﬁne the automorphism Eij : xi → xi xj . 3) For any i ∈ {1, 2, . . . , n} we deﬁne the automorphism Ni : xi → x−1 i . We denote Nij = Ni Nj for i = j. The kernel of the canonical epimorphism Aut(Fn ) → GLn (Z) is denoted by IA(Fn ). It is known, that IA(F2 ) = Inn(F2 ) and IA(Fn ) is strictly larger than Inn(Fn ) for n 3. J. Nielsen for n 3 [17] and W. Magnus for all n [12] proved that IA(Fn ) is generated by all automorphisms αijk (see also [11]).

4. Finite index subgroups of Aut(Fn) containing IA(Fn) Theorem 4.1. Let n 3. Any subgroup of ﬁnite index in Aut(Fn ), containing IA(Fn ), has a ﬁnite abelianization. To prove this theorem we need to introduce more automorphisms of Aut(Fn ) and to formulate a theorem of B. Sury and T.N. Venkataramana (see below) on generators of congruence subgroups of SLn (Z). 4) For any i ∈ {1, 2, . . . , n} we deﬁne the automorphism −1 Ti : xi → x−1 i , xi+1 → xi+1 xi .

5) For any i, j ∈ {1, 2, . . . , n}, where i = j, we deﬁne the automorphism Tij : xi → xj , xj → x−1 i . Denote T = {Tk | k = 1, . . . , n − 1} ∪ {Tij | i, j = 1, 2, . . . , n; i = j} ∪ {I}, where I is the identity automorphism of Fn . Let ¯ : Aut(Fn ) → GLn (Z) be the canonical epimorphism. For any α ∈ Aut(Fn ) we denote by α its canonical image in GLn (Z).

8

O. Bogopolski and R. Vikentiev

Let m be a natural number. The kernel of the canonical epimorphism SLn (Z) → SLn (Zm ) is denoted by SLn (Z, mZ) and is called the congruence subgroup of SLn (Z) modulo m. Of course, SLn (Z, mZ) is normal and has a ﬁnite index in SLn (Z). The following theorem is called the congruence subgroup theorem for SLn (Z). It was proved by H. Bass, M. Lazard and J.-P. Serre [1] and independently by J. Mennicke [16]. Theorem 4.2. Let n 3 be a natural number. Any subgroup of ﬁnite index in SLn (Z) contains a congruence subgroup SLn (Z, mZ) for some m. Theorem 4.3. (B. Sury and T.N. Venkataramana; 1994). Let n 3, m 2. The congruence subgroup SLn (Z, mZ) is generated by the following set of matrices {(α)(E ij )m (α)−1 | α ∈ T ; i, j ∈ {1, 2, . . . , n}; i = j}. Corollary 4.4. The group SLn (Z, 2Z), n 3, is generated by the set 2

X = {E ij , N ij | i, j ∈ {1, 2, . . . , n}; i = j}. 2

Proof. By Theorem 4.3, SLn (Z, 2Z) is the normal closure of all E ij in SLn (Z). Since SLn (Z) is generated by all transvections E pq , it is suﬃcient to verify that

−

for every x ∈ X and ∈ {−1, 1} holds E pq xE pq ∈ X. We leave this to the reader. Proof of Theorem 4.1. Let G be a subgroup of ﬁnite index in Aut(Fn ) containing IA(Fn ). We show that G/G is ﬁnite. Let G be the image of G is GLn (Z). The index of G ∩ SLn (Z) in SLn (Z) is ﬁnite. Hence, by the congruence subgroup theorem, there exists an m 2, such that SLn (Z, mZ) G. Let H be the full preimage of SLn (Z, mZ) in G. Since the index of H in G is ﬁnite, it is suﬃcient to show that H/H is ﬁnite. Since H contains IA(Fn ), the results of Nielsen–Magnus and Sury–Venkataramana imply that H is generated by the union of two sets: {αijk | i, j, k ∈ {1, 2, . . . , n}; k = i, j}, m −1 α | α ∈ T ; i, j ∈ {1, 2, . . . , n}; i = j}. {αEij

It is suﬃcient to prove that the mth power of each of these generators lies in [H, H]. But this follows from the following formulas: m 1) [αiij , Ejk ] = αm iik for distinct i, j, k ∈ {1, 2, . . . , n}; m m−1 αikj αm−1 ≡ αm 2) [αiij , Eik ] = (αikj α−1 ikj mod[IA(Fn ), IA(Fn )] for disjjk ) jjk tinct i, j, k ∈ {1, 2, . . . , n}; m2 −(s+1)m m2 −(s+1)m m−2 s s+1 m2 m m = [α , E ][E , α ] [Eij , Ejk ]. 3) Eik iij iij s=0 ik ik

5. Congruence subgroups SCong(n, k) in SAut(Fn) Let G be a group and H be a normal subgroup in G. We denote Aut(G; H) = {ϕ ∈ Aut(G) | ϕ(H) = H}.

Subgroups of Small Index in Aut(Fn )

9

and IA(G; H) = {ϕ ∈ Aut(G) | ∀g ∈ G : ϕ(gH) = gH}. Equivalently IA(G; H) = {ϕ ∈ Aut(G) | ∀g ∈ G ∃xg ∈ H : ϕ(g) = g · xg }.

(1)

Clearly, IA(G; H) is normal in Aut(G; H) and the corresponding factor group is naturally embeddable into Aut(G/H). In particular, if |G : H| = 2, then we have IA(G; H) = Aut(G; H). Proposition 5.1. Let {Hi | i ∈ I} be a set of normal subgroups of a group G. Then IA G; ∩ Hi = ∩ IA G; Hi . i∈I

i∈I

Proof. The proof is straightforward from description (1).

Now we return to automorphisms of Fn = F (x1 , . . . , xn ). Let k 2 be a natural number. Consider the standard epimorphisms π

ε

Fn −→ Zn −→ Znk . They induce the epimorphisms π

ε

Aut(Fn ) −→ Aut(Zn ) −→ Aut(Znk ). Using usual identiﬁcations we may write π

ε

Aut(Fn ) −→ GLn (Z) −→ GLn (Zk ). We denote Cong(n; k) = Ker(π ), SCong(n; k) = Cong(n; k) ∩ SAut(Fn ) and call these groups the congruence subgroups of Aut(Fn ) and of SAut(Fn ) modulo k, respectively. In this paper we will work only with congruence subgroups modulo 2. Let {σi | i = 1, . . . , 2n − 1} be the set of all epimorphisms Fn → Z2 . The (i) kernel of σi is a free group of rank 2n − 1; we denote it by F2n−1 . We ﬁx σ1 by (1) the rule σ1 (x1 ) = · · · = σ1 (xn−1 ) = 0 and σ1 (xn ) = 1. Thus, F2n−1 has the basis 2 −1 −1 {x1 , . . . , xn−1 , xn , xn x1 xn , . . . , xn xn−1 xn }. For brevity we denote (i)

(i)

St(F2n−1 ) = Aut(Fn , F2n−1 ). Proposition 5.2. For n 3 holds: 1) Aut(Fn )/Cong(n, 2) ∼ = GLn (Z2 ), 2) |Cong(n, 2) : SCong(n, 2)| = 2, 2n −1 (i) St(F2n−1 ), 3) Cong(n, 2) = i=1

10

O. Bogopolski and R. Vikentiev

4) SCong(n, 2) is generated by the set X = {αijk | i, j, k ∈ {1, 2, . . . , n}, k = i, j}

2 , Nij | i, j ∈ {1, 2, . . . , n}, i = j}. {Eij

5) The abelianization of SCong(n, 2) is a ﬁnite 2-group. Proof. 1) This statement follows from the deﬁnition of the congruence subgroup. 2) We deﬁne the automorphism ϕ ∈ Aut(Fn ) by the rule x1 → x1 x22 , x2 → x1 x22 x1 x2 , xi → xi for i = 1, . . . , n. It is easy to see, that ϕ ∈ Cong(n, 2) \ SCong(n, 2). 3) This statement follows from the chain of identities (with k = 2): −1 −1 2 2 IA Fn ; Ker(σi ) = Aut Fn ; Ker(σi ) . Ker(π ε) = IA Fn ; Ker(πε) = n

n

i=1

i=1

The ﬁrst identity is evident, the second one follows from Proposition 5.1 and the 2n −1 fact that Ker(πε) = Ker(σi ). The third identity follows from the fact that i=1

|Fn : Ker (σi )| = 2. 4) This statement follows from Corollary 4.4 and Nielsen–Magnus result. 5) The abelianization of SCong(n, 2) is ﬁnite by Theorem 4.1. The following computations (where i, j, k are distinct) show, that it is a ﬁnite 2-group: 2 [αiij , Ejk ] = α2iik , 2 2 [αiij , Eik ] = αikj α−1 jjk αikj αjjk ≡ αikj mod IA(Fn ) , 2 4 [Eij , Njk ] = Eij ,

Nij2 = 1.

Remark 5.3. Using ﬁrst Johnson homomorphism and some homological methods, T. Satoh proved in [19], that for n 2 and k 2 holds Cong(n, k) ∼ = IA(Fn ) ⊗Z Zk ⊕ Γ(n, k) , where Γ(n, k) is the congruence subgroup of GLn (Z) modulo k.

6. A subgroup K(n) of index 2 in SCong(n, 2) We have the following chain of canonical embeddings and epimorphisms. θ1

(1)

θ2

θ

SCong(n, 2) → St(F2n−1 ) → Aut(F2n−1 ) →3 GL2n−1 (Z). Here θ1 is the embedding due to Proposition 5.2.3, θ2 is the homomorphism, (1) (1) which sends every automorphism α of Fn with α(F2n−1 ) = F2n−1 to its restriction (1) on F2n−1 . In fact, θ2 is injective. The epimorphism θ3 is standard. Now we set θ = θ1 θ2 θ3 and deﬁne the subgroup K(n) = θ−1 SL2n−1 (Z) .

Subgroups of Small Index in Aut(Fn )

11

Proposition 6.1. For n 3 holds 1) 2) 3) 4)

|SCong(n, 2) : K(n)| = 2, |IA(Fn ) : IA(Fn ) ∩ K(n)| = 2, |Inn(Fn ) : Inn(Fn ) ∩ K(n)| is 1 if n is odd, and is 2 if n is even, K(n) is generated by the set Y = X \ {αiin , Nin | i = 1, 2, . . . , n − 1} ∪ {N1n αiin | i = 1, 2, . . . , n − 1}, where

X = {αijk | i, j, k ∈ {1, 2, . . . , n}, k = i, j}

2 , Nij | i, j ∈ {1, 2, . . . , n}, i = j} {Eij

is the generating set for SCong(n, 2) deﬁned in Proposition 5.2. Proof. 1) Clearly, |SCong(n, 2) : K(n)| 2. One can check that N1n ∈ SCong(n, 2) \ K(n). Therefore this index is indeed 2. 2) Since IA(Fn ) SCong(n, 2), we have by 1), that |IA(Fn ) : IA(Fn ) ∩ K(n)| 2. One can verify, that α11n ∈ IA(Fn ) \ K(n). Therefore this index is indeed 2. 3) By 2), this index does not exceed 2. For x ∈ Fn let x ˆ be the conjugation of Fn by x, i.e., x ˆ(y) = x−1 yx for y ∈ Fn . To prove the statement, it is suﬃcient to check that x ˆ1 , . . . , xˆn−1 ∈ K(n) and that x ˆn ∈ K(n) if and only if n is odd. 4) We take {1, N1n } as the set of representatives of the cosets of K(n) in SCong(n, 2). For g ∈ SCong(n, 2), we denote by g the representative of the coset K(n)g. One can easily check, that for g ∈ X holds N1n , if g ∈ {Nin , αiin | i = 1, . . . , n − 1} g= 1, if g ∈ X \ {Nin , αiin | i = 1, . . . , n − 1}. By the Reidemeister–Schreier method, K(n) is generated by the elements −1

N1n xN1n x , where x runs through X. We show, that these elements can be expressed as products of elements of Y ±1 . Consider the following cases. I. Let x = αiin , where i ∈ {1, . . . , n − 1}. −1

Then N1n xN1n x

= N1n αiin ∈ Y .

II. Let x = αijk , where i, j, k ∈ {1, . . . , n} are distinct or x = αiik with k = n (the case x = αiik with k = n was considered above). −1

−1 Then N1n xN1n x = N1n αijk N1n . We consider several cases and rewrite this element as a product of elements of Y ±1 . We will use the fact that αijk = α−1 ikj if i, j, k are distinct.

Case 1. {1, n} ⊆ {i, j, k}. Subcase 1.1. αijk is of the ﬁrst kind, i.e., i, j, k are diﬀerent.

12

O. Bogopolski and R. Vikentiev a) i = 1. Then n ∈ {j, k} and we may assume that n = j. Then −1 −2 −1 2 N1n α1nk N1n = N1k α−1 11k E1n α1kn α11k E1n N1k .

b) i = n. Then 1 ∈ {j, k} and we may assume that j = 1. Then −1 −1 −1 −1 = α−1 N1n αn1k N1n n1k αnnk αnn1 αnnk αnn1 αkk1 αn1k αkk1 αn1k .

c) i = 1, n. Then {1, n} = {j, k} and we may assume that j = 1 and k = n. Then −1 −2 −1 2 N1n αi1n N1n = Ein αii1 αin1 Ein αii1 .

Subcase 1.2. αijk is of the second kind, i.e., it is αiik . By our assumption in II we have k = n. Therefore i = n and k = 1, and we have −1 N1n αnn1 N1n = α−1 nn1 .

Case 2. n ∈ {i, j, k}, 1 ∈ / {i, j, k}. Then we choose t ∈ {i, j, k} \ {n} and write −1 −1 −1 = N1t (Ntn αijk Ntn )N1t . N1n αijk N1n

The expression in the brackets can be computed as in Case 1 (by replacing 1 by t). Case 3. n ∈ / {i, j, k}, 1 ∈ {i, j, k}. Subcase 3.1. αijk is of the ﬁrst kind, i.e., i, j, k are diﬀerent. a) i = 1. Then −1 −1 N1n α1jk N1n = α1kj · α11k α11j α−1 11k α11j .

b) j = 1. Then −1 = α−1 N1n αi1k N1n ii1 αik1 αii1 .

c) k = 1. This case reduces to Case b), since αijk = α−1 ikj . Subcase 3.2. αijk is of the second kind, i.e., it is αiik . a) i = 1. Then −1 = α11k . N1n α11k N1n

b) k = 1. Then −1 N1n αii1 N1n = α−1 ii1 .

Case 4. {1, n} ∩ {i, j, k} = ∅. −1 N1n αijk N1n = αijk .

Subgroups of Small Index in Aut(Fn ) 2 . III. Let x = Eij

13

−1

−1 2 Then N1n xN1n x = N1n Eij N1n . The following formulas show, that this element belongs to Y . −1 −2 2 N1n = α2nnj Enj N1n Enj

(j = 1, n),

−1 −2 2 N1n E1j N1n = α211j E1j

(j = 1, n),

−1 2 N1n N1n Ein −1 2 N1n Ei1 N1n −1 2 N1n En1 N1n −1 2 N1n E1n N1n

IV. Let x = Nij . If i, j = n, then N1n xN1n x −1

N1n xN1n x

= = = = −1

−2 Ein (i = 1, n), −2 Ei1 (i = 1, n), 2 α−2 nn1 En1 , −2 −1 N12 E1n N12 .

= Nij . If, say j = n, then i = n and

= N1i .

7. K(3) and some its overgroups with inﬁnite abelianization The congruence subgroup SCong(3, 2) has index 2 · 168 in Aut(F3 ) and contains IA(F3 ). Therefore, by Theorem 4.1, it has a ﬁnite abelianization. Moreover, by Proposition 5.2, this abelianization is a ﬁnite 2-group. The subgroup K(3) has index 2 in SCong(3, 2) and does not contain IA(F3 ) by Proposition 6.1. This was an indication for us to check that the abelianization of K(3) is inﬁnite. In this section we also construct some overgroups of K(3), namely A(3), B(3) and C(3), with inﬁnite abelianizations. In computing the abelianizations of these groups, we used their generators, Nielsen’s presentation of Aut(F3 ) (see [18] or [13]) and the Reidemeister–Schreier method implemented in the GAP package. By Proposition 6.1 we have the following 16 Generators of K(3): (1) α112 , α221 , α331 , α332 , α123 , α213 , α312 , 2 2 2 2 2 2 (2) E12 , E13 , E21 , E23 , E31 , E32 , (3) N13 α113 , N13 α223 , N12 . Theorem 7.1. 1) K(3) has index 4 · 168 in Aut(F3 ). 2) K(3)/K(3) ∼ = Z14 2 × Z × Z. Corollary 7.2. The above 16 automorphisms form a minimal generating set of 2 2 , E21 have ﬁnite order in K(3)/K(3) . K(3). All of them, except of α123 , α213 , E12 These four automorphisms have inﬁnite order in K(3)/K(3) . −4 −4 ≡ α2123 (mod K(3) ) and E21 ≡ α2213 (mod K(3) ). In particuMoreover, E12 2 2 lar, the image of the group E12 , E21 in K(3)/K(3) is isomorphic to Z × Z.

14

O. Bogopolski and R. Vikentiev

Proof. The ﬁrst statement of this corollary follows straightforward from Theorem 7.1. The second statement follows from the formulas [α221 , N12 ] = α2221 , [α112 , N12 ] = α2112 , 2 ] = α2332 , [α331 , E12 2 [α332 , E21 ] = α2331 , −2 2 2 [α331 , E32 ] = α321 α−1 112 α321 α112 ≡ α321 = α312 (mod K(3) ), 2 4 [E31 , N12 ] = E31 , 2 4 [E32 , N12 ] = E32 , 2 2 4 [α221 , E13 ][E23 , N12 ] = E23 , 2 2 4 ][E13 , N12 ] = E13 , [α112 , E23

(N13 α113 )2 = 1, (N13 α223 )2 = 1, 2 N12 = 1.

This statement and Theorem 7.1.2) imply that the image of 2 2 , E21

α123 , α213 , E12

in K(3)/K(3)

can be mapped onto Z × Z. Therefore all remaining statements of the corollary −4 −4 will follow, if we prove the congruences E12 ≡ α2123 (mod K(3) ) and E21 ≡ 2 α213 (mod K(3) ). The ﬁrst congruence follows from the identity −1 −2 −1 2 −4 −2 −4 2 ] = α332 α−1 [α113 N23 , E12 123 α332 α112 α123 α112 E12 ≡ α123 E12 (mod K(3) ) 2 and the fact that the commutator [α113 N23 , E12 ] belongs to K(3) ; indeed, we have 2 α113 N23 ∈ K(3) and E12 ∈ K(3). The second congruence follows similarly if we exchange indices 1 and 2.

Still, the index of K(3) in Aut(F3 ) is large. We will enlarge K(3) (and so decrease the index) by adding special generators. In this way we construct the following chain of subgroups: Aut(F3 ) C(3) B(3) A(3) K(3), where A(3) = K(3), E31 , E32 , B(3) = K(3), E31 , E32 , E21 , C(3) = K(3), E31 , E32 , E21 , N3 .

Subgroups of Small Index in Aut(Fn )

15

Theorem 7.3. 1) A(3) has index 168 in Aut(F3 ). 2) A(3)/A(3) ∼ = Z72 × Z × Z. 3) A(3) has the following minimal set of generators: 2 2 α112 , α221 , α123 , α213 , E13 , E23 , E31 , E32 , N12 .

Theorem 7.4. 1) B(3) has index 84 in Aut(F3 ). 2) B(3)/B(3) ∼ = Z42 × Z. Remark. We do not know a minimal generating set of B(3). Theorem 7.5. 1) C(3) has index 42 in Aut(F3 ). 2) C(3)/C(3) ∼ = Z32 × Z4 × Z. 3) C(3) has the following minimal set of generators: 2 α123 , E13 , E21 , E32 , N3 .

Questions 7.6. Does there exist a subgroup of Aut(F3 ) of index smaller than 42, which can be mapped onto Z?

8. The group K(n) for n 4 Theorem 8.1. If n 4, then the abelianization of K(n) is a ﬁnite 2-group. Proof. By Proposition 6.1, K(n) is generated by the set Y = X \ {αiin , Nin | i = 1, 2, . . . , n − 1} ∪ {N1n αiin | i = 1, 2, . . . , n − 1}, where

X = {αijk | i, j, k ∈ {1, 2, . . . , n}, k = i, j}

2 , Nij | i, j ∈ {1, 2, . . . , n}, i = j}. {Eij We show, that the square of each y ∈ Y is trivial modulo K(n) . 4 1. We show that Eij ≡ 1 mod K(n) . For j = n, this follows from the identity −2 −4 , Njk ] = Eij , [Eij

where we choose k ∈ / {i, j, n}. For j = n, this follows from the identity 2 2 4 [αiik , Ekn ][Ein , Nik ] = Ein ,

where we choose k ∈ / {i, n}. 2. We show that α2iij ≡ 1 mod K(n) . This follows from the identity 2 [αiik , Ekj ] = α2iij ,

16

O. Bogopolski and R. Vikentiev

/ K(n), but where we choose k ∈ / {i, j, n}. Note an interesting fact, that αiin ∈ α2iin ∈ K(n) . 3. We show that α2ijk ≡ 1 mod K(n) for diﬀerent i, j, k. If n ∈ / {j, k}, then this follows from 2 2 ] = αijk α−1 [αiik , Eij kkj αijk αkkj ≡ αijk mod K(n)

If n ∈ {j, k}, we may assume that k = n and then the required congruence follows from 2 2 ] = αijn αiij αijn α−1 [αiin Nln , Eij iij ≡ αijn mod K(n) , where we take l ∈ / {i, j, n}. Note, that in this case αiin Nln lies in K(n). 4. One can easily check, that (N1n αiin )2 = 1 holds for all i = 1, . . . , n − 1. Finally, it is clear that Nij2 = 1. Remark 8.2. Using the GAP package we have proved that the abelianization of K(4) is isomorphic to Z38 2 . Questions 8.3. Is it true that K(n) /K(n) is inﬁnite for n 4? If this is true, then Aut(Fn ) has no the Kazhdan property (T) for n 4.

References [1] H. Bass, M. Lazard, J.-P. Serre. Sous-groupes d’indice ﬁni dans SL(n, Z), Bull. Amer. Math. Soc., 70 (1964), 385–392. [2] B. Bekka, P. de la Harpe, A. Valette, Kazhdan’s property (T ), New Mathematical Monographs, 11, Cambridge: Cambridge University Press, 2008. [3] O. Bogopolski, Arboreal decomposability of groups of automorphisms of free groups, Algebra and Logic, 26, no. 2 (1987), 79–91. [4] O. Bogopolski, Classiﬁcation of automorphisms of the free group of rank 2 by ranks of ﬁxed-points subgroups, J. Group Theory, 3, no. 3 (2000), 339–351. [5] F. Grunewald, A. Lubotzky, Linear representations of the automorphism group of a free group, GAFA, 18, no. 5 (2009), 1564–1608. [6] P. de la Harpe, A. Valette, La propri´et´e (T) de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger), Ast´erisque 175, 1989. [7] S. Hoory, N. Linial, A. Wigderson, Expander graphs and their applications, Bulletin of the Amer. Math. Soc., 43, no. 4 (2006), 439–561. [8] D. Kazhdan, On the connection of the dual space of a group with the structure of its closed subgroups, Functional analysis and its applications, 1 (1967), 63–65. [9] A. Lubotzky, Discrete groups, expanding graphs and invariant measures, BaselBoston-Berlin: Birkh¨ auser, 1994. [10] A. Lubotzky, I. Pak, The product replacement algorithm and Kazhdan’s property (T), J. Amer. Math. Soc., 14, no. 2 (2001), 347–363. [11] R.C. Lyndon, P.E. Schupp, Combinatorial group theory, Berlin: Springer-Verlag, 1977.

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¨ [12] W. Magnus, Uber n-dimensionale Gittertransformationen, Acta Math., 64 (1935), 353–367, [13] W. Magnus, A. Karrass, D. Solitar, Combinatorial group theory, New York: Wiley, 1966. [14] G.A. Margulis, Explicit constructions of concentrators. Problems of Inform. Transm., 10 (1975), 325–332. [Russian original: Problemy Peredatci Informacii, 9 (1973), 71– 80.] [15] J. McCool, A faithful polynomial presentation of Out(F3 ), Math. Proc. Camb. Phil. Soc., 106, no. 2 (1989), 207–213. [16] J. Mennicke, Finite factor groups of the unimodular group, Ann. Math., Ser 2., 81 (1965), 31–37. [17] J. Nielsen, Die Gruppe der dreidimensionalen Gittertransformationen, Danske Vid. Selsk. Mat.-Fys. Medd. 12 (1924), 1–29. [18] J. Nielsen, Die Isomorphismengruppe der freien Gruppen, Math. Ann, 91 (1924), 169–209. [19] T. Satoh, The abelianization of the congruence IA-automorphism group of a free group, Math. Proc. Camb. Philos. Soc., 142, no. 2 (2007), 239–248. [20] J.-P. Serre, Trees. Berlin–Heidelberg–New York: Springer-Verlag, 1980. [21] Y. Shalom, The algebraization of Kazhdan’s property (T). Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures. Zurich: European Mathematical Society (EMS). 1283– 1310 (2006). [22] B. Sury, T.N. Venkataramana, Generators for all principal congruence subgroups of SLn (Z) with n > 2, Proc. Amer. Math. Soc., 122, no. 2 (1994), 355–358. [23] K. Vogtmann, Automorphisms of free groups and outer space, Geometriae Dedicata, 94 (2002), 1–31. [24] Y. Watatani, Property (T) of Kazhdan implies property (FA) of Serre, Math. Japon., 27, no. 1 (1982), 97–103. O. Bogopolski Institute of Mathematics of Siberian Branch of Russian Academy of Sciences Novosibirsk, Russia and University of D¨ usseldorf, Germany e-mail: Oleg [email protected] R. Vikentiev Institute of Mathematics of Siberian Branch of Russian Academy of Sciences Novosibirsk, Russia and Novosibirsk State University e-mail: [email protected]

Combinatorial and Geometric Group Theory Trends in Mathematics, 19–53 c 2010 Springer Basel AG

Dynamics of Free Group Automorphisms Peter Brinkmann Abstract. We present a coarse convexity result for the dynamics of free group automorphisms: Given an automorphism φ of a ﬁnitely generated free group F , we show that for all x ∈ F and 0 ≤ i ≤ N , the length of φi (x) is bounded above by a constant multiple of the sum of the lengths of x and φN (x), with the constant depending only on φ. Mathematics Subject Classiﬁcation (2000). 37E30. Keywords. Free group automorphisms, train tracks.

Introduction The following theorem is the main result of this paper. It follows from a technical result (Theorem 1.9) that uses the machinery of improved relative train track maps of Bestvina, Feighn, and Handel [BFH00]. Theorem 0.1. Let φ : F → F be an automorphism of a ﬁnitely generated free group. Then there exists a constant K ≥ 1 such that for any pair of exponents N, i satisfying 0 ≤ i ≤ N , the following two statements hold: 1. If w is a cyclic word in G, then

||φi# (w)|| ≤ K ||w|| + ||φN # (w)|| ,

where ||w|| is the length of the cyclic reduction of w with respect to some word metric on F . 2. If w is a word in F , then |φi# (w)| ≤ K |w| + |φN (w)| , where |w| is the length of w. Given an improved relative train track representative of some power of φ, the constant K can be computed. Remark 0.2 (A note on computability). Given an automorphism φ : F → F, we can compute a relative train track representative of φ [BH92, DV96]. The construction

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of improved relative train track maps, however, involves a compactness argument in a universal cover [BFH00, Proof of Proposition 5.4.3] that is not constructive. A number of algorithmic improvements of relative train tracks appear in [Bri], in the context of an algorithm that detects automorphic orbits in free groups. The statement of the theorem does not depend on the choice of generators of F . The intuitive meaning of the theorem is that the map i → |φi (w)| is coarsely convex for all words w ∈ F . Klaus Johannson informed me that a similar result is a folk theorem in the case of surface homeomorphisms. Also, while free-by-cyclic groups are not, in general, CAT(0)-groups [Ger94], Theorem 0.1 suggests that their dynamics mimics that of CAT(0)-groups. Theorem 0.1 complements the following strong convexity result in an important special case. Theorem 0.3 ([Bri00]). If φ : F → F is an atoroidal automorphism, i.e., φ has no nontrivial periodic conjugacy classes, then φ is hyperbolic, i.e., there exists a constant λ > 1 such that λ|x| ≤ max |φ±1 (x)| for all x ∈ F . I originally set out to prove Theorem 0.1 because it immediately implies that in a free-by-cyclic group Γ = F φ Z = x1 , . . . , xn , t | t−1 xi t = φ(xi ) , words of the form t−k wtk φk (w−1 ) satisfy a quadratic isoperimetric inequality. (Note, however, that Theorem 0.1 is stronger than the mere existence of a quadratic isoperimetric inequality for such words.) Natasa Macura previously proved a quadratic isoperimetric inequality for mapping tori of automorphisms of polynomial growth [Mac00]. Martin Bridson and Daniel Groves have since proved that all free-by-cyclic groups satisfy a quadratic isoperimetric inequality [BG]. They also obtain a new proof of Theorem 0.1 as an application of their techniques. In Section 1, we review the pertinent deﬁnitions and results regarding train track maps from [BFH00]. We also state the main technical result, Theorem 1.9, and we show how Theorem 0.1 follows from Theorem 1.9. Section 2 provides some more results on train tracks and automorphisms of free groups. Section 3 introduces some notation and terminology and lists a number of examples that illustrate some of the issues and subtleties that need to be addressed in the proof of Theorem 1.9. Section 4 establishes a technical proposition that may be of independent interest. Section 5 and Section 6 contain the proof of Theorem 1.9. Finally, the glossary lists some of the technical deﬁnitions for the convenience of the reader. I would like to express my gratitude to Ilya Kapovich for many helpful discussions, to Mladen Bestvina for patiently answering my questions, to Steve Gersten for encouraging me to write up this result for its own sake, to the University of Osnabr¨ uck for their hospitality, and to Swarup Gadde and the University of Melbourne as well as the Max-Planck-Institute of Mathematics for their hospitality and ﬁnancial support. Klaus Johannson and Richard Weidmann kindly served as a sounding board while I was working on the exposition of this paper.

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1. Improved relative train track maps In this section, we review the theory of train tracks developed in [BH92, BFH00]. We will restrict our attention to the collection of those results that we will use in this paper. Given an automorphism φ ∈ Aut(F ), we can ﬁnd a based homotopy equivalence f : G → G of a ﬁnite connected graph G such that π1 (G) ∼ = F and f induces φ. This observation allows us to apply topological techniques to automorphisms of free groups. In many cases, it is convenient to work with outer automorphisms. Topologically, this means that we work with homotopy equivalences rather that based homotopy equivalences. Oftentimes, a homotopy equivalence f : G → G will respect a ﬁltration of G, i.e., there exist subgraphs G0 = ∅ ⊂ G1 ⊂ · · · ⊂ Gk = G such that for each ﬁltration element Gr , the restriction of f to Gr is a homotopy equivalence of Gr . The subgraph Hr = Gr \ Gr−1 is called the rth stratum of the ﬁltration. We say that a path ρ has nontrivial intersection with a stratum Hr if ρ crosses at least one edge in Hr . If E1 , . . . , Em is the collection of edges in some stratum Hr , the transition matrix of Hr is the nonnegative m × m-matrix Mr whose ijth entry is the number of times the f -image of Ej crosses Ei , regardless of orientation. Mr is said to be irreducible if for every tuple 1 ≤ i, j ≤ m, there exists some exponent n > 0 such that the ijth entry of Mrn is nonzero. If Mr is irreducible, then it has a maximal real eigenvalue λr ≥ 1 [Gan59]. We call λr the growth rate of Hr . Given a homotopy equivalence f : G → G, we can always ﬁnd a ﬁltration of G such that each transition matrix is either a zero matrix or irreducible. A stratum Hr in such a ﬁltration is called zero stratum if Mr = 0. Hr is called exponentially growing if Mr is irreducible with λr > 1, and it is called polynomially growing if Mr is irreducible with λr = 1. An unordered pair of edges in G originating from the same vertex is called a turn. A turn is called degenerate if the two edges are equal. We deﬁne a map Df : {turns in G} → {turns in G} by sending each edge in a turn to the ﬁrst edge in its image under f . A turn is called illegal if its image under some iterate of Df is degenerate, legal otherwise. An edge path ρ = E1 E2 · · · Es is said to contain the turns (Ei−1 , Ei+1 ) for 1 ≤ i < s. ρ is said to be legal if all its turns are legal, and a path ρ ⊂ Gr is r-legal if no illegal turn in α involves an edge in Hr . Let ρ be a path in G. In general, the composition f k ◦ ρ is not an immersion, but there is exactly one immersion that is homotopic to f k ◦ ρ relative endpoints. k k We denote this immersion by f# (ρ), and we say that we obtain f# (ρ) from f k ◦ ρ k by tightening. If σ is a circuit in G, then f# (σ) is the immersed circuit homotopic to f k ◦ σ. Remark 1.1. A path is tightened by cancelling adjacent pairs of inverse edges until no inverse pairs are left. The result of such a sequence of cancellations is uniquely

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determined, but the sequence is not. For instance, EE −1 E may be tightened as E(E −1 E) or (EE −1 )E. Convention 1.2. Let ρi , i = 1, . . . , k be paths that can be concatenated to form a path ρ = ρ1 ρ2 · · · ρk . When tightening f (ρ) to obtain f# (ρ), we adopt the convention that we ﬁrst tighten the images of ρi to f# (ρi ). In a second step, we tighten the concatenation f# (ρ1 ) · · · f# (ρk ) to f# (ρ). In many situations, the length of a subpath ρi will be greater than the number of edges that cancel at either end, in which case it makes sense to talk about edges in f# (ρ) originating from ρi . k (ρ) = ρ for some k > 0. In this A path ρ is a (periodic) Nielsen path if f# case, the smallest such k is the period of ρ. A Nielsen path ρ is called indivisible if it cannot be expressed as the concatenation of shorter Nielsen paths. A path ρ is k a pre-Nielsen path if f# (ρ) is Nielsen for some k ≥ 0. A decomposition of a path ρ = ρ1 · ρ2 · · · · · ρs into subpaths is called a kk k k splitting if f# (ρ) = f# (ρ1 ) · · · f# (ρs ). Such a decomposition is a splitting if it is a k-splitting for all k > 0. We will also use the notion of k-splittings of circuits σ = ρ1 · ρ2 · · · · · ρs , which requires, in addition, that there be no cancellation k k between f# (ρs ) and f# (ρ1 ). The following theorem was proved in [BH92].

Theorem 1.3 ([BH92, Theorem 5.12]). Every outer automorphism O of F is represented by a homotopy equivalence f : G → G such that each exponentially growing stratum Hr has the following properties: 1. If E is an edge in Hr , then the ﬁrst and last edges in f (E) are contained in Hr . 2. If β is a nontrivial path in Gr−1 with endpoints in Gr−1 ∩ Hr , then f# (β) is nontrivial. 3. If ρ is an r-legal path, then f# (ρ) is an r-legal path. We call f a relative train track map. A path ρ in G is said to be of height r if ρ ⊂ Gr and ρ ⊂ Gr−1 . If Hr = {Er } is a polynomially growing stratum, then basic paths of height r are of the form Er γ or Er γEr−1 , where γ is a path in Gr−1 . If τ is a closed Nielsen path in Gr−1 and f (Er ) = Er τ l for some l ∈ Z, then paths of the form Er τ k and Er τ k Er−1 are exceptional paths of height r. Moreover, if s < r, τ ⊂ Gs−1 , and f (Es ) = Es τ m , then Er τ k Es−1 is also a exceptional path of height r. For our purposes, the properties of relative train track maps are not strong enough, so we will use the notion of improved train track maps constructed in [BFH00]. We only list the properties used in this paper. Theorem 1.4 ([BFH00, Theorem 5.1.5, Lemma 5.1.7, and Proposition 5.4.3]). For every outer automorphism O of F , there exists an exponent k > 0 such that Ok is represented by a relative train track map f : G → G with the following additional properties:

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1. If Hr is a zero stratum, then Hr+1 is an exponentially growing stratum, and the restriction of f to Hr is an immersion. Hr is a zero stratum if and only if it is the union of the contractible components of Gr . 2. If v is a vertex, then f (v) is a ﬁxed vertex. If Hr is a polynomially growing stratum and G is the collection of noncontractible components of Gr−1 , then all vertices in Hr ∩ G are ﬁxed. 3. If Hr is an exponentially growing stratum, then there is at most one indivisible Nielsen path τ of height r. If τ is not closed and if it starts and ends at vertices, then at least one endpoint of τ is not contained in Hr ∩ Gr−1 . 4. If Hr is a polynomially growing stratum, then Hr consists of a single edge Er , and f (Er ) = Er · ur for some closed path ur ⊂ Gr−1 whose base point is ﬁxed by f . If σ ⊂ Gr is a basic path of height r that does not split as a concatenation of two basic paths of height r or as a concatenation of a basic path of height r k with a path contained in Gr−1 , then either f# (σ) = Er · σ for some k ≥ 0, k or ur is a Nielsen path and f# (σ) is an exceptional path of height r for some k ≥ 0. We call f an improved relative train track map. Finally, we state a lemma from [BFH00] that simpliﬁes the study of paths intersecting strata of polynomial growth. Lemma 1.5 ([BFH00, Lemma 4.1.4]). Let f : G → G be an improved train track map with a polynomially growing stratum Hr . If ρ is a path in Gr , then it splits as a concatenation of basic paths of height r and paths in Gr−1 . Remark 1.6. In fact, part 4 of Theorem 1.4 implies that subdividing ρ at the initial endpoints of all occurrences of Er and at the terminal endpoints of all occurrences of Er−1 yields a splitting of ρ into basic paths of height r and paths in Gr−1 . k Observe that if Hr = {Er } is a polynomially growing stratum, then f# (Er ) = k−1 i Er · ur · f# (ur ) · · · · · f# (ur ). Each subpath of the form f# (ur ) is called a block k of f# (Er ). Since there is no cancellation between successive blocks, it makes sense to refer to the inﬁnite path 2 (ur ) · · · · Rr = ur · f# (ur ) · f#

(1)

as the eigenray of Er . Remark 1.7 (A note on terminology). The notion of a polynomially growing stratum Hr = {Er } ﬁrst appeared in [BH92]. Polynomially growing strata are called nonexponentially growing strata in [BFH00]. Both terms are somewhat misleading k because the function k → |f# (Er )| may grow exponentially (see Lemma 2.4). Given an improved train track map f : G → G, we construct a metric on G. If Hr is an exponentially growing stratum, then its transition matrix Mr has a unique positive left eigenvector vr (corresponding to λr ) whose smallest entry

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equals one [Gan59]. For an edge Ei in Hr , the eigenvector vr has an entry li > 0 corresponding to Ei . We choose a metric on G such that Ei is isometric to an interval of length li , and such that edges in zero strata or in polynomially growing strata are isometric to an interval of length one. For a path ρ, we denote its length by L(ρ). Note that if the endpoints of ρ are vertices, then the number of edges in ρ provides a lower bound for L(ρ). Moreover, if f is an absolute train track map, then f expands the length of legal paths by the factor λ. Remark 1.8. We merely choose this metric for convenience. All statements here are invariant under bi-Lipschitz maps, but our metric of choice simpliﬁes the presentation of our arguments. We are now ready to state the main technical result of this paper. Theorem 1.9. Let φ : F → F be an an automorphism. Then there exists an improved relative train track map representing some positive power of φ for which there exists a constant K ≥ 1 with the following property: For any pair of exponents N, i satisfying 0 ≤ i ≤ N , the following two statements hold: 1. If σ is a circuit in G, then i N L f# (σ) ≤ K L(σ) + L f# (σ) . 2. If ρ is a path in G that starts and ends at vertices, then N i (ρ) ≤ K L(ρ) + L f# (ρ) . L f# Given the improved relative train track map f : G → G, the constant K can be computed. We will present the proof of Theorem 1.9 in Section 5 and Section 6. Right now, we show how Theorem 0.1 follows from Theorem 1.9. Proof of Theorem 0.1. Let φ : F → F be an automorphism of a ﬁnitely generated free group F = x1 , . . . , xn . The ﬁrst part of Theorem 1.9 immediately implies that the ﬁrst part of Theorem 0.1 holds for some positive power φk , i.e., there exists some K ≥ 1 such that for all 0 ≤ i ≤ N and w ∈ F , we have Nk ||φik # (w)|| ≤ K ||w|| + ||φ# (w)|| , where we compute lengths with respect to the generators x1 , . . . , xn . Let L = max{|φ±1 (xi )|}. Then, for 0 ≤ j < k, we have L−k ||φik+j (w)|| ≤ ||φik (w)|| ≤ Lk ||φik+j (w)|| for all w ∈ F . We conclude that for all 0 ≤ i ≤ N and w ∈ F , we have L−k ||φi# (w)|| ≤ K ||w|| + Lk ||φN # (w)|| , so that the ﬁrst part of Theorem 0.1 holds with K = L2k K . In order to prove the second assertion, we modify a trick from [BFH97]. Let F be the free group generated by x1 , . . . , xn and an additional generator a. We

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deﬁne an automorphism ψ : F → F by letting ψ(xi ) = φ(xi ) for all 1 ≤ i ≤ n, and ψ(a) = a. By the previous step, the ﬁrst part of Theorem 0.1 holds for ψ, with some constant K ≥ 1. Let w be some word in F . Then, for all i ≥ 0, ψ i (aw) is a cyclically reduced word in F , so that we have |φi (w)| + 1 = ||ψ i (aw)||. We conclude that |φi (w)| + 1 ≤ K (|w| + |φN (w)| + 2), for all 0 ≤ i ≤ N . Now the second assertion of Theorem 0.1 holds with K = 2K .

2. More on train tracks Thurston’s bounded cancellation lemma is one of the fundamental tools in this paper. We state it in terms of homotopy equivalences of graphs. Lemma 2.1 (Bounded cancellation lemma [Coo87]). Let f : G → G be a homotopy equivalence. There exists a constant Cf , depending only on f , with the property that for any tight path ρ in G obtained by concatenating two paths α, β, we have L(f# (ρ)) ≥ L(f# (α)) + L(f# (β)) − Cf . An upper bound for Cf can easily be read oﬀ from the map f [Coo87]. Let f : G → G be an improved relative train track map with an exponentially growing stratum Hr with growth rate λr . The r-length of a path ρ in G, Lr (ρ), is the total length of ρ ∩ Hr . If β is an r-legal path in G whose r-length satisﬁes λr Lr (β) − 2Cf > Lr (β) and α, γ are paths such that the concatenation αβγ is an immersion, then the k r-length of the segment in f# (αβγ) corresponding to β (Convention (1.2)) will tend to inﬁnity as k tends to inﬁnity. The critical length Cr of Hr is the inﬁmum of the lengths satisfying the above inequality, i.e., 2Cf . (2) Cr = λr − 1 We now list some additional technical results about improved train track maps. The following lemma is an immediate consequence of [Bri00, Prop. 6.2]. If Hr is an exponentially growing stratum, and ρ is a path of height r, we let n(ρ) denote the number of r-legal segments in ρ. Lemma 2.2. Let f : G → G be a relative train track map, and let Hr be an exponentially growing stratum. For each L > 0, there exists some computable exponent M > 0 such that if ρ is a path or circuit in Gr containing at least one full edge in Hr , one of the following three statements holds: M 1. f# (ρ) has an r-legal segment of r-length greater than L. M (ρ)) < n(ρ). 2. n(f# 3. ρ can be expressed as a concatenation τ1 ρ τ2 , where τ1 , τ2 each contain at most one r-illegal turn, the r-length of the r-legal segments of τ1 , τ2 is at most L,

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P. Brinkmann and ρ splits as a concatenation of pre-Nielsen paths (with one r-illegal turn M each) and segments in Gr−1 . Moreover, f# (ρ ) is a concatenation of Nielsen paths of height r and segments in Gr−1 .

Remark 2.3. • The statement of Lemma 2.2 in [Bri00] does not explicitly mention the computability of M . The proof, however, only uses counting arguments, from which the constant M can be computed. • The presence of the subpaths τ1 , τ2 in Part 2.2 is an artifact of the fact that ρ need not start or end at ﬁxed points if it is a path. If ρ starts at a ﬁxed point, then τ1 will be trivial, and if ρ ends at a ﬁxed point, then τ2 will be trivial. • The actual statement of [Bri00, Proposition 6.2] does not mention circuits since they were not a concern in the context of [Bri00]. The proof, however, works for circuits as well as paths. If the ﬁrst two statements of Lemma 2.2 do not hold, than the third statement will hold with τ1 and τ2 trivial. From now on, we assume that f : G → G that f is an improved train track map. Throughout the rest of this section, let M be the constant from Lemma 2.2 for some ﬁxed L > Cr (Equation 2). Let Hr = {Er } be a polynomially growing stratum. We say that Hr is truly polynomial if ur is trivial or, inductively, if ur is a concatenation of truly polynomial edges and Nielsen paths in exponentially growing strata. Clearly, Er is truly k polynomial if and only if the map k → |f# (Er )| grows polynomially. We say that a polynomially growing stratum is fast if it is not truly polynomial. The following lemma give us an understanding of the growth of fast polynomial strata. Lemma 2.4. There exists an exponent k0 with the following property: For all fast polynomial strata Hr = {Er } there exists some s < r such that Hs is of expok0 nential growth and f# (Er ) contains an s-legal subpath of height s whose s-length exceeds Cs . In particular, this lemma implies that fast polynomial strata grow exponentially. Given an improved relative train track map, we can ﬁnd k0 by successively 2 evaluating f# , f# , . . . until we see long legal segments in all images of fast polynomial edges. Proof. We introduce classes of fast polynomial edges. Let Hr = {Er } be a fast polynomial edge such that f (Er ) = Er ur . We say that Hr has class 1 if there exists some s < r such that Hs is an exponentially growing stratum, ur ∩ Hs does not only consist of Nielsen paths and paths of height less than s, and if ur contains any polynomial edges Et for some t > s, then Et is truly polynomial. We recursively deﬁne a fast polynomial edge Er to have class k if the highest class of edges in ur is k − 1.

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k (ρ) If Hr has class 1, then ur contains a subpath ρ of height s such that f# contains a long s-legal segment for some suﬃciently large k (Lemma 2.2). If ur contains any subpaths whose height exceeds s, then by deﬁnition those subpaths will grow at most polynomially, so that eventually, the exponential growth of ρ will prevail. In order to prove the lemma for an edge of class k, k > 1, we observe that no m (ur ). Now the edges of class k − 1 are cancelled when f m (ur ) is tightened to f# lemma follows by Theorem 1.4, Part 4, and induction.

Assume that Hr is an exponentially growing stratum, and let ρ be a path of height r. If Hr does not support a closed Nielsen path, then we let N (ρ) = n(ρ). If Hr supports a closed Nielsen path, then we let N (ρ) equal the number of legal segments in ρ that do not overlap with a closed Nielsen subpath of ρ. The following lemma is a generalization of [Bri00, Lemma 6.4]. Lemma 2.5. Assume that Hr is an exponentially growing stratum. There exist M computable constants λ > 1, N0 with the following property: If f# (ρ) does not contain a legal segment of length at least L, and if N (ρ) > N0 , then M N (f# (ρ)) ≤ λ−1 N (ρ).

Regardless of N (ρ), we have M N (f# (ρ)) ≤ λ−1 N (ρ) + 1.

Proof. If Hr does not support a closed Nielsen path, then the proof of [Bri00, Lemma 6.4] goes through unchanged. We repeat the argument here because the ideas of the proof show up more clearly in this case. If Hr does not support a closed Nielsen path, then the proof is based on the following observation: If N (ρ) = 6 and f# (ρ) does not contain a long legal M M segment, then N (f# (ρ)) < 6. Suppose otherwise, i.e., N (f# (ρ)) = N (ρ). Then, by M Lemma 2.2, f# (ρ) = τ1 γτ2 , where γ is a concatenation of three indivisible Nielsen paths of height r and paths in Gr−1 . This is impossible because by Theorem 1.4, Part 3, we can concatenate no more than two indivisible Nielsen paths of height r with paths in Gr−1 . Hence, of every six consecutive legal segments in ρ, at least one cancels comM (ρ). This implies that if N (ρ) ≥ 6, then pletely when f M (ρ) is tightened to f# 10 M N (f# (ρ)) ≤ 11 N (ρ). In order to see why this choice of λ works, we just observe M that if ρ consists of eleven legal segments and the sixth one cancels in f# (ρ), then M there are no six consecutive legal segments that survive in f# (ρ). This completes the proof of the ﬁrst inequality, with λ = 11 and N0 = 5, if 10 Hr does not support a closed Nielsen path. Regarding the second inequality, we M remark that if N (ρ) ≤ N0 , then N (f# (ρ)) ≤ N (ρ) ≤ λ−1 N (ρ) + 1. We now assume that Hr supports a closed indivisible Nielsen path σ. The proof in this case is based on the following consequence of Lemma 2.2. If γ a path of M M height r, n(γ) = n(f# (γ)) = 4, and f# (γ) does not contain a long legal segment,

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M (γ) = τ1 σ ±1 τ2 , where τ1 and τ2 are as in Lemma 2.2. Intuitively, this then f# means that if few legal segments disappear, then many Nielsen paths will appear. Since N (ρ) only counts those legal segments that do not overlap with a Nielsen path, this observation will yield the desired estimate. First, consider a path γ of height r that does not contain any Nielsen subpaths, i.e., we have N (γ) = n(γ). If N (γ) ≥ 4, then for every four consecutive M M legal segments whose images do not cancel completely in f# (γ), f# (γ) contains 6 M at least one Nielsen subpath, so that we have N (f# (γ)) ≤ 7 N (γ), using the same reasoning as above. We claim that if γ starts and ends at ﬁxed points, then, by Remark 2.3, we M have N (f# (γ)) ≤ 67 N (γ) regardless of N (γ). To this end, we ﬁrst argue that if γ M (γ)) < n(γ). If this were not true, then, starts and ends at ﬁxed points, then n(f# M m by Lemma 2.2 we would have f# (γ) = σ for some m ∈ Z, which would imply that γ = σ m because γ starts and ends at ﬁxed points. This is a contradiction since we assumed that γ does not contain any Nielsen subpaths. Now, if n(γ) = N (γ) < 4, M M then we conclude that N (f# (γ)) ≤ n(f# (γ)) < n(γ). Now n(γ) < 4 implies that 6 6 6 M n(γ) ≥ n(γ) − 1, which implies that N (f # (γ)) ≤ 7 n(γ) = 7 N (γ). 7 After these preparations, we express ρ as a concatenation

ρ = ρ1 σ n1 ρ2 σ n2 ρ3 · · · ρk σ nk ρk+1 , where n1 , . . . , nk ∈ Z, and none of the subpaths ρi contains a Nielsen subpath. Note that the subpaths ρ2 , . . . , ρk start and end at the base point v of the M Nielsen path σ, which is ﬁxed by f . Hence, for 2 ≤ i ≤ k, we have N (f# (ρi )) ≤ 6 6 6 M M 7 N (ρi ), and we have N (f# (ρ1 )) ≤ 7 N (ρ1 ) (resp. N (f# (ρk+1 )) ≤ 7 N (ρk+1 )) if N (ρ1 ) ≥ 4 (resp. N (ρk+1 ) ≥ 4). If N (ρ1 ) < 4 and N (ρk+1 ) < 4, we have 6 N (ρ1 ) + N (ρk+1 ) + (N (ρ) − N (ρ1 ) − N (ρk+1 )) 7 6 6 ≤ 6 + (N (ρ) − 6) ≤ (1 + N (ρ)). 7 7 6 M Similar estimates yield that N (f# (ρ)) ≤ 7 (1 + N (ρ)) regardless of N (ρ1 ) and N (ρk+1 ). M If N (ρ) > 11, then 67 (1 + N (ρ)) ≤ 13 14 N (ρ), which implies that N (f# (ρ)) ≤ 13 N (ρ) if N (ρ) > 11, so that the ﬁrst inequality of the lemma holds with λ = 14 14 13 M and N0 = 11. As for the second inequality, we remark that N (f# (ρ)) ≤ N (ρ) and, if N (ρ) ≤ 11, then N (ρ) ≤ λ−1 N (ρ) + 1. M (ρ)) ≤ N (f#

The next lemma is a statement about the (absence of) cancellation between eigenrays of polynomially growing strata. It is a stronger version of [BFH00, Sublemma 1, Page 587]. Lemma 2.6. Let Hi = {Ei } and Hj = {Ej } be polynomially growing strata. Let Si (resp. Sj ) be an initial segment of Ei Ri (resp. Ej Rj , see Equation 1) such that

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the concatenation Si S¯j is a path. If Ei grows faster than linearly and if an entire k block of Rj is cancelled in f# (Si S¯j ) for some k ≥ 0, then no entire block of Ri l will be cancelled in f# (Si S¯j ) for any l ≥ 0. Proof. Suppose that at least one block of both Si and Sj cancels. Then there are k l (ui ) = βγ, f# (uj ) = αβ for some k, l ≥ 0, and paths α, β, and γ such that f# f# (α) = γ (see Figure 1). Ei k (ui ) f#

α

β β

γ γ

l f# (uj )

Ej Figure 1. The idea of the proof of Lemma 2.6. In particular, we have k−1 2 Ri = ui f# (ui ) · · · f# (ui )βf# (α)f# (β)f# (α) . . . ,

and l−1 2 (uj )αβf# (α)f# (β)f# (α) . . . . Rj = uj f# (uj ) · · · f#

¯ l−1 E ¯j does not split. By Theorem 1.4, ρ is In particular, the path ρ = Ei Rik−1 α ¯R j a exceptional path, and both Ei and Ej grow linearly.

3. Terminology and examples In this section, we discuss some examples that illustrate some of the main issues that we need to address in the proof of Theorem 1.9. Although we are not primarily concerned with free-by-cyclic groups in this article, the language of free-by-cyclic groups will streamline the exposition. Given a free group Fn = x1 , . . . , xn and an automorphism φ of Fn , the mapping torus of φ is the free-by-cyclic group Mφ = x1 , . . . , xn , t | t−1 xi tφ(x−1 i ). The letter t is called the stable letter of Mφ . Deﬁnition 3.1. A reduced word w in the generators of Mφ is a hallway if w represents the trivial element of Mφ and if w can be expressed as w = w1 w2 such that w1 only contains negative powers of t and w2 only contains positive powers of t [BF92]. Hallways of the form t−k xtk φk (x−1 ), for x ∈ Fn , are said to be smooth.

30

P. Brinkmann Any hallway w can be expressed as w = t−1 uk−1 t−1 uk−2 t−1 · · · t−1 u1 t−1 w0 tv1 tv2 t · · · tvk−1 twk−1 ,

where w0 , wk , u1 , . . . , uk−1 , v1 , . . . , vk−1 are elements of Fn . The words ui and vi may be empty. In fact, a hallway is smooth if and only if all the ui and vi are trivial. For 1 ≤ i < k, we deﬁne wi to be the word obtained by tightening ui φ(wi−1 )vi . Since w represents the identity, we have wk = φ(wk−1 ). We call wi the ith slice of w. The number k is the duration D(w) of the hallway. Figure 2 illustrates these notions. uk−1 u1 wk

wi

w0

k vk−1 v1 v2 Figure 2. A hallway.

We say that the instances of letters of Fn that occur in the spelling of w are visible. Theorem 0.1 states that if w is a smooth hallway, then the length of each wi is bounded by a constant multiple of the number of visible edges in w. The following examples illustrate the main issues that arise in the proof. For the remainder of this section, let F6 = a, b, c, d, x, y, and deﬁne φ by letting a

→

a

b c

→ →

ba caa

d x

→ dc → y

y

→

xcy.

This automorphism admits the stratiﬁcation H1 = {a}, H2 = {b}, H3 = {c}, H4 = {d}, and H5 = {x, y}. The restriction of f to the ﬁltration element G3 = H1 ∪ H2 ∪ H3 grows linearly, the restriction to G4 grows quadratically, and the stratum H5 is of exponential growth.

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31

The ﬁrst example illustrates the behavior of smooth hallways in linearly growing ﬁltration elements. Example 3.2. Let w0 be a word from the list am , bam b−1 , cam c−1 , for some integer m. Then φ(w0 ) = w0 , so that the length of any slice of the hallway t−k w0 tk w0−1 is the same as the length of w0 . Now, let w0 be a word from the list bam , cam , cam b−1 . If m ≥ 0, then |φk+1 (w0 )| = |φk (w0 )|+1 for any k ≥ 0. If m < 0, then |φk+1 (w0 )| = |φk (w0 )| − 1 for 0 ≤ k < −m (Figure 3). Hence, the length of each slice of the hallway t−k w0 tk φk (w0−1 ) is bounded by the number of visible letters. b b a3 a

2

c

c

Figure 3. Illustration of Example 3.2. If w0 is an arbitrary word in a, b, c, then, by Remark 1.6, it splits as a concatenation of words from the above lists and their inverses, which implies that the lengths of slices of smooth hallways is bounded by the number of visible letters, so that Theorem 0.1 holds with K = 1. The next example shows that hallways that are not smooth may have slices whose length is not bounded in terms of a constant multiple of the number of visible edges. Example 3.3. Let w = t−k ct−k b−1 t2k bc−1 . For i < k, we have wi = a−i b−1 , and for k ≤ i ≤ 2k, we have wi = ca2k−i b−1 (Figure 4). In particular, there is a slice of length k + 2 although there are only four visible edges in w. Informally, one might say that hallways of this form bulge in the middle. A similar bulge occurs for hallways of the form w = t−k b−1 t−k btk b−1 tk b. The next example shows that we need to control the size of such bulges when proving Theorem 1.9. Example 3.4. First, note that for k ≥ 1, the last letter in the words f −k (xc) is always one of x, y, x−1 , y −1, so that words of the form w0 = φ−k (xc)b−1 are reduced, and we have φk (w0 ) = xcak b−1 and φ2k (w0 ) = φk (x)b−1 . Hence, the smooth hallway w = t−2k w0 t2k φ2k (w0−1 ) contains a bulge like the ﬁrst one in the previous example (Figure 5). The presence of this bulge does not

32

P. Brinkmann c

b

b

ak

ak c b

b

b

b

Figure 4. Illustration of Example 3.3.

x c

φk (x)

φ−k (xc) ak

c b

b

Figure 5. Illustration of Example 3.4. contradict Theorem 0.1 because w contains a large number of visible instances of the letters x and y. This example shows that we cannot consider the strata separately when proving Theorem 1.9. Example 3.5. If we let w0 = φ−k (dc)b−1 , then the smooth hallway w = t−2k w0 t2k φ2k (w0−1 ) contains a bulge like in Example 3.4. This does not contradict Theorem 0.1 as w contains a large number of visible instances of the letter c. Our ﬁnal example illustrates a subtlety regarding linearly growing strata. Example 3.6. Let F4 = a, b, c, d and deﬁne ψ by letting a → b →

a ba

→

ca

c

d →

dcb−1 .

Dynamics of Free Group Automorphisms

33

The map ψ is a linearly growing automorphism, so in particular the letter d is of linear growth, although the image of d contains letters of linear growth other than d itself. Letters of linear growth may thus behave in two diﬀerent ways; they may contribute to the growth of images under successive applications of ψ, or they may remain inert as parts of a ﬁxed word. In the proof of Theorem 1.9, we will need to distinguish letters of linear growth according to their role. Example 3.7. Let F3 = a, x, y and deﬁne φ by letting a

→

axyx−1 y −1

x → y −1 y → yx. The stratum {x, y} grows exponentially, and we have ψ(xyx−1 y −1 ) = xyx−1 y −1 . This means that a grows linearly although it maps across an exponentially growing stratum. This is another phenomenon that we need to consider when analyzing strata of linear growth. The notion of hallways naturally extends to mapping tori of homotopy equivalences of ﬁnite graphs. Speciﬁcally, a hallway ρ in the mapping torus of f : G → G is a sequence of paths of the form ρ = (μk−1 , μk−2 , . . . , μ1 , ρ0 , ν1 , ν2 , . . . , νk−1 , ρk ), where ρ0 , ρk , μ1 , . . . , μk−1 , ν1 , . . . , νk−1 are paths in G, satisfying f (τ (ρ0 )) = ι(ν1 ), f (τ (νi )) = ι(νi+1 ), f (τ (νk−1 )) = τ (ρk ), f (ι(ρ0 )) = τ (μ1 ), f (ι(μi )) = τ (μi+1 ), and f (ι(μk−1 )) = ι(ρk ), where ι(.) is the initial point of a path, and τ (.) is the terminal point. The paths μi and νi are called notches. Some or all of the notches may be trivial. For 1 ≤ i < k, we deﬁne ρi to be the path obtained by tightening μi f (ρi−1 )νi . Since ρ is a closed path, we have ρk = f# (ρk−1 ). As before, we call ρi the ith slice of ρ, and the number k is the duration D(ρ). Deﬁnition 3.8. The visible length of ρ is V(ρ) = L(ρ0 ) + L(ρk ) +

k−1

(L(μi ) + L(νi )) .

i=1

Finally, we introduce quasi-smooth hallways: Given some C ≥ 0, we say that w hallway ρ is C-quasi-smooth if the length of all the notches is bounded by C.

4. Strata of superlinear growth Throughout this section, let f : G → G be an improved relative train track map. In order to track images of edges through the slices of a hallway ρ, we assign a marking to each edge. This assignment will, in general, involve arbitrary choices, but our arguments will not be aﬀected by these choices.

34

P. Brinkmann

Deﬁnition 4.1. We begin by marking all edges in the initial slice ρ0 and in all notches μi , νi with their height. Assume inductively that all edges in a slice ρi−1 have been marked, and let E be an edge of height r in ρi−1 , with marking s. Now, consider an edge E in f (E). If the height of E is r, or if Hs is a zero stratum, then we keep the marking s. If the height of E is less than r, then we mark E by r. This gives us a marking for all edges in μi f (ρi−1 )νi . Note that, as we tighten μi f (ρi−1 )νi to obtain ρi , diﬀerent choices in cancellation (Remark 1.1) may give rise to diﬀerent possible markings, but this will not be a problem. We say that an edge E is marked by a linear/polynomial/exponential stratum if its marking is s and Hs is linear/polynomial/exponential. The following proposition goes a long way toward proving Theorem 1.9. Proposition 4.2. There exists some constant K ≥ 1 such that for every hallway ρ and every slice ρi of ρ, the number of edges in ρi that are not marked by strata of linear growth is bounded by KV(ρ). Given the improved relative train track map f : G → G, the constant K can be computed. If f has no edges of linear growth, then Proposition 4.2 immediately implies Theorem 1.9: For a smooth hallway ρ of duration k, we have V(ρ) = L(ρ0 ) + k i L(ρk ) = L(ρ0 ) + L(f# (ρ0 )) and ρi = f# (ρ0 ), so that Proposition 4.2 implies k L(ρi ) ≤ K(L(ρ0 ) + L(f# (ρ0 )) in this case. In order to streamline the exposition, we will not always make the choice of K explicit. However, it will turn out that K can be chosen to be the product of numbers that can easily be read oﬀ from the train track map. The intuition of the proof is that once signiﬁcant growth occurs, it will be due to the presence of long legal subpaths in exponentially growing strata or long subsegments of eigenrays of polynomially growing strata that grow faster than linearly. Lemma 2.1 and Lemma 2.6 imply that there is hardly any cancellation between such subpaths and their surroundings, so that any signiﬁcant growth that occurs in a slice will eventually be accounted for by visible edges. The following deﬁnition will help us understand cancellation in hallways. For every stratum Hr , we deﬁne a number h(Hr ) in the following way: • If Hr is a constant stratum, then h(Hr ) = 0. • If Hr is a nonconstant polynomially growing stratum, i.e., Hr = {Er } and f (Er ) = Er ur , then h(Hr ) is the height of ur . • If Hr is of exponential growth and Hr−1 is not a zero stratum, then h(Er ) is the height of f (Hr ) ∩ Gr−1 , unless this intersection does not contain any edges, in which case we let h(Hr ) = ∞. • If Hr is of exponential growth and Hr−1 is a zero stratum, then h(Er ) is the height of f (Hr ∪ Hr−1 ) ∩ Gr−2 . We also let h(Hr−1 ) = h(Hr ).

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35

Essentially, h(Hr ) is the index of the highest stratum crossed by the image of Hr , other than Hr itself. We may permute the strata of G (while preserving the improved train track properties) such that h(Hr ) > h(Hs ) implies r > s. Given a stratum Hs , we say that the set S(Hs ) = {Hr |h(Hr ) = s} is the league of Hs , the motivation being that they, in a sense, “play at the same level.” If h(Hr ) = ∞, then Hr does not belong to any league. Proof of P roposition 4.2. First of all, we note that if a slice ρi has a subpath in a zero stratum Hr , then this subpath is of uniformly bounded length, and it is surrounded by edges in higher strata (Theorem 1.4, Part 1), so that we have a linear estimate of the number of edges in Hr in ρi in terms of the number of edges in higher strata. Let q be the largest (ﬁnite) number for which the league S(q) is nonempty. Fix some stratum Hr for r > q. We want to ﬁnd a linear bound on the number of edges in ρi ∩ Hr in terms of visible edges. By deﬁnition of S(q) and choice of r, edges in ρi ∩ Hr never cancel with edges from other strata or their images. If Hr = {Er } is of polynomial growth, then any occurrence of Er in ρi is the image of a visible copy of Er , and ρi contains at most one copy of Er for each visible copy of Er . Hence, the number of edges in ρi ∩ Hr is bounded by the number of visible edges. Now, assume that Hr is an exponentially growing stratum. A slice ρi decomposes into r-legal subpaths with r-illegal turns in between. By Lemma 2.1, a subpath whose r-length is greater than Cr (Equation 2) will eventually be accounted for by visible edges since it will not be shortened by cancellation within slices. Edges in Hr whose r-distance from an illegal turn is less than C2r may cancel eventually, and ρi contains at most Cr of them per r-illegal turn, so that we only need to ﬁnd a bound of the number of r-illegal turns in terms of the number of visible edges. Since the improved train track map f does not create any r-illegal turns, any r-illegal turn in ρi can be traced back to a visible illegal turn in ρ (or an illegal turn created by appending a notch to the image of a slice). This implies that the number of r-illegal turns in ρi is bounded by the number of visible edges in ρ. Summing up, we have bounded the number of edges in ρi ∩(Hq+1 ∪Hq+2 ∪· · · ) by a multiple of the number of visible edges. This establishes the base case of the proof. We now assume inductively that the number of edges in S(p) ∪ S(p + 1) ∪ · · · has been bounded as a constant multiple of V(ρ). We need to ﬁnd a bound on the number of edges in ρi ∩ Hp . We ﬁrst assume that Hp = {Ep } is of polynomial growth. By deﬁnition of S(p), an edge in ρi ∩ Hp has one of four possible markings: • Its marking may be p, indicating that it is the image of a visible edge, or • it may be marked by an exponentially growing stratum in S(q), for some q ≥ p, or

36

P. Brinkmann • it may be marked by a superlinear polynomially growing stratum in S(q), q ≥ p, or • it may be marked by a stratum of linear growth.

We are not concerned with edges of the fourth kind. As before, the number of edges of the ﬁrst kind in ρi ∩ Hp is bounded by the number of visible edges. Let C be the largest number of copies of Ep that occur in the image of a single edge in an exponentially growing stratum Hs , for s > p. Then the number of edges of the second kind in ρi ∩ Hp is bounded by C times the number of exponentially growing edges in ρi−1 ∩ (S(p) ∩ S(p + 1) ∩ · · · ), which in turn is bounded by a multiple of the number of visible edges. We have no immediate bound on the number of edges of the third kind. As we trace the image of such an edge through subsequent slices, one of three possible events will occur: • Either, it eventually maps to a visible edge, or • it cancels with an edge of the ﬁrst or second kind, or • it cancels with an edge in the image of a polynomially growing (possibly linearly growing) edge in S(p). Note that these events may depend on choices in tightening (Remark 1.1), but once again our estimates will not be aﬀected by these choices. The number of edges for which one of the ﬁrst two events occurs is clearly bounded by a multiple of the number of visible edges. We only need to ﬁnd a bound on the number of edges in an eigenray that eventually cancel with edges in another eigenray. Lemma 2.6 implies that there is a uniform bound on the number of edges in Hp that cancel when two rays meet, so that we only need to ﬁnd a bound on the number of meetings between two rays. Clearly, any two rays meet at most once. If an eigenray cancels with segments from more than one other ray (this is conceivable since a slice may be of the form ρi = Er S1 S2 , where Er is a polynomially growing edge in S(p) and S1 , S2 are short segments from rays of edges in S(p) such that the ray of Er successively cancels with S1 and S2 ), then all except possibly one of these segments cancel completely, so that they are no longer available for subsequent cancellation. This implies that the number of meetings of rays is bounded by two times the number of pieces of rays available for cancellation, which in turn is bounded by the number of visible edges. This completes our estimate of the number of edges in ρi ∩ Hp when Hp is of polynomial growth. We now assume that Hp is of exponential growth. The number of subpaths of height p of ρi is bounded by the number of edges of height greater than p in ρi plus one. The contribution of p-legal subpaths of p-length less than or equal to Cp is bounded by Cp times the number of subpaths of height p, so that we do not need to consider them here. Any p-legal subpaths of length greater than Cp will eventually show up in the visible part of ρ, so that we do not need to consider them, either. The remaining edges in ρi ∩ Hp are at

Dynamics of Free Group Automorphisms

37

C

p-distance less than 2p from a p-illegal turn. Hence, we only need to ﬁnd a bound on the number of p-illegal turns in ρi . As before, we trace illegal turns in ρi back to their origin: • An illegal turn may be the image of a visible illegal turn (this case includes illegal turns created by appending notches μi , νi to the image f# (ρi−1 ) of a slice), or • it may come from a illegal turn in the image of an exponentially growing edge in S(p), or • it may be contained in the ray of a polynomially growing edge in S(p), or • it may be contained in a Nielsen path marked by a linear stratum (Example 3.7). We are not concerned with illegal turns of the fourth type. The same arguments that we used for polynomially growing Hp yield that the number of illegal turns of the ﬁrst and second kind is bounded by a multiple of the number of visible edges. Now, let C be the maximum of the number of illegal turns in the images of polynomially growing edges in S(p). Lemma 2.4 yields an exponent k0 such that k0 for polynomially growing edge Er in S(p), f# (ur ) contains a long legal segment. k This means, in particular, that if ρ contains a block f# (ur ), k ≥ k0 , then this block contains no more than C illegal turns per long legal segment. Since long legal segments eventually show up as visible edges, the number of illegal turns in such blocks is bounded by CV(ρ). The remaining illegal turns are contained in initial subpaths of rays that contain no more than the ﬁrst C + 1 blocks, i.e., there are at most C(C + 1) illegal turns of this kind per ray. Since we already know that the number of rays is bounded in terms of the number of visible edges, we are done in this case. We have now obtained the desired estimate for edges in ρi of height p and higher. In particular, this includes all strata in S(p − 1), which completes the inductive step.

5. Polynomially growing automorphisms In this section, we establish Theorem 1.9 in the case of polynomially growing automorphisms. Speciﬁcally, we ﬁnd estimates for the contribution of linearly growing edges that we ignored in Proposition 4.2. As usual, let f : G → G be an improved relative train track map. Since f is of polynomial growth, every stratum Hr contains only one edge Er , and we have f (Er ) = Er · ur , where ur is some closed path in Gr−1 . Note that all vertices of G are ﬁxed. We ﬁrst record an obvious lemma. Lemma 5.1. Let μ1 , μ2 be Nielsen paths in G, and let ν be some path in G. • If μ1 and μ2 can be concatenated, then the path obtained from μ1 μ2 by tightening relative endpoints is also a Nielsen path.

38

P. Brinkmann • If μ1 and ν can be concatenated, let γ be the path obtained by tightening μ1 ν, and let Δ = L(γ) − L(ν). Then, for all k ≥ 0, we have k k (γ) = L f# (ν) + Δ and − L(μ1 ) ≤ Δ ≤ L(μ1 ). L f#

We now establish Theorem 1.9 for automorphisms of linear growth. This lemma will provide the base case of our inductive proof of Theorem 1.9. Lemma 5.2. Assume that f : G → G is of linear growth. If ρ is a smooth hallway, and if ρ0 starts and ends at vertices, then the lengths of slices of ρ are bounded by V(ρ), i.e., T heorem 1.9 holds with K = 1. Proof. The proof proceeds by induction up through the strata of G. The bottom stratum H1 is constant, so that the lemma trivially holds for the restriction of f to H1 . We now assume that Hr is a linearly growing stratum, and that the lemma holds for the restriction of f to Gp−1 . Consider the initial slice ρ0 . Remark 1.6 yields a splitting of ρ0 into basic paths of height p and paths in Gp−1 . The splitting of ρ0 induces a decomposition of ρ into smooth hallways, so that it suﬃces to prove the claim for hallways whose initial slice is a basic path of height p or a path in Gp−1 . By induction, we only need to prove the claim if ρ0 is a basic path of height p. If the basic path ρ0 is, in fact, an exceptional path, then the reasoning of Example 3.2 proves our claim, so that we may assume that ρ0 is not an exceptional path. Assume that ρ0 is a basic path of the form Ep γ. Then, by Theorem 1.4, m+1 (Ep γ) splits as Part 4, there exists some smallest exponent m ≥ 0 for which f# Ep · γ . Using Remark 1.6 once more, we conclude that Ep γ can be expressed as Ep u−m p ν. · ν. We can consider the subpaths If D(ρ) ≤ m, then ρ0 k-splits as Ep u−m p −m Ep up and ν separately, so that we are done in this case. i f# (ν) Now assume that D(ρ) > m. For 0 ≤ i ≤ m, we have ρi = Ep ui−m p i and L(ρi ) = 1 + (m − i)L(up ) + L f# (ν) . For m + 1 ≤ i ≤ D(ρ), we have i−(m+1) i i ρi = Ep up f# (up ν) and L(ρi ) = 1 + (i − m − 1)L(up ) + L f# (ν) + Δ, where Δ is deﬁned as in Lemma 5.1. D(ρ) We have V(ρ) = 2+(D(ρ) − 1) L(up )+L(ν)+L f# (ν) +Δ. By induction, D(ρ) i we have L f# (ν) ≤ L(ν) + L f# (ν) for all 0 ≤ i ≤ D(ρ). This immediately implies that L(ρi ) ≤ V(ρ) for all 0 ≤ i ≤ D(ρ). If ρ0 = Ep γEp−1 , we essentially repeat the same argument. Once more, we can write ρ0 = Ep u−m p ν, and in order to use the previous argument, we only need to know that the lemma holds for ν. This, however, follows from the previous step, so that we are done. We now ﬁnd estimates on the number of edges emitted by linearly growing edges, the quantity we ignored in Proposition 4.2. The idea is to take a hallway

Dynamics of Free Group Automorphisms

39

and decompose it into smaller and smaller pieces until all remaining pieces only involve linearly growing edges and their rays. Simple counting arguments will give us bounds on the number of the remaining pieces as well as the lengths of their slices. Let ρ be a hallway, and assume that there is a visible edge Er that does not cancel within ρ, i.e., we can trace its image through the slices of ρ until it reappears as another visible edge. Then ρ can be expressed as ρ = αEr βEr−1 , and we deﬁne two new hallways ρ , ρ by tightening t−k Er βEr−1 and αtk . We say that ρ and ρ are obtained from ρ by cutting along the trajectory of Er (Figure 6). The exponent k is the length of the cut. We say that a hallway ρ is indecomposable if it does not admit any cuts of length D(ρ). cut α

sawtooth construction t

Er

k

Er Er

Er ur

β Figure 6. Cutting and the sawtooth construction. Now we obtain a new hallway σ from ρ by repeatedly replacing subwords of the form t−1 Er by f (Er )t−1 and tightening (Figure 6). We refer to this operation as the sawtooth construction along the trajectory of Er . If M is a collection of hallways, we let V(M) = V(σ). σ∈M

The following lemma lists some basic properties of our two operations. We k (E) grows polynomially of degree d. say that an edge is of degree d if f# Lemma 5.3. Fix some C ≥ max{L(ur )}. Let ρ be a C-quasi-smooth hallway in G. Choose d > 1 such that the fastest growing edge crossed by ρ grows polynomially of degree d. Obtain a collection M of hallways by cutting along all trajectories of edges E in ρ of degree d. Let M1 be the collection of smooth elements of M, and let M2 consist of hallways obtained by performing the sawtooth construction along all trajectories of E of degree d in those elements of M that are not smooth. Then 1. The duration of all elements of M1 and M2 is at most D(ρ). 2. None of the elements of M2 crosses edges of degree d, i.e., they only cross edges of degree at most d − 1. 3. All elements of M2 are 2C-quasi-smooth.

40

P. Brinkmann

4. The number of elements of M2 is bounded by 2CD(ρ). 5. We have 2 V(M1 ) + V(M2 ) ≤ V(ρ) + (2CD(ρ)) . Proof. The ﬁrst four properties follow immediately from deﬁnitions. In order to prove the ﬁfth property, we just remark that each element of M2 has at most 2CD(ρ) visible edges that do not appear in ρ itself. Since M2 contains at most 2CD(ρ) hallways, the estimate follows. Lemma 5.4. There exists a (computable) constant C with the following property: Let γ be a path of height r, starting and ending at vertices, and assume that Er is of degree d > 1. Then, for all k ≥ 0, k L(γ) + L(f# (γ)) ≥ Ck d .

Proof. It suﬃces to prove the lemma if either γ = Er γ , or γ = Er γ Es−1 , where γ only involves edges of degree less than d, and Es is of degree d. In the ﬁrst case, the claim is obvious. In the second case, we remark that Lemma 2.6 guarantees that there is hardly any cancellation between the rays of Er and Es , so that the lemma follows. The following proposition implies the second part of Theorem 1.9 in the case of polynomially growing automorphisms. In particular, it provides bounds on the number of edges emitted by linearly growing edges. This is the quantity that we ignored in Proposition 4.2. Proposition 5.5. Assume that f represents an automorphism that grows polynomially of degree q. Fix some C ≥ max{L(ur )}. There exist computable constants K1 ≤ K2 ≤ · · · ≤ Kq and K1 (C), . . . , Kq (C) such that 1. If ρ is a smooth hallway whose fastest growing edge is of degree d, and if ρ0 starts and ends at vertices, then L(ρi ) ≤ Kd V(ρ) for all slices ρi of ρ. 2. If ρ is a C-quasi-smooth hallway whose fastest growing edge is of degree d, then in every slice ρi , the number of edges emitted by linearly growing edges is bounded by Kd V(ρ) + Kd (C)Dd+1 (ρ), so that we have L(ρi ) ≤ (K + Kd )V(ρ) + Kd (C)Dd+1 (ρ), where K is the constant from P roposition 4.2. Proof. We prove the proposition by induction on d. For d = 1, the ﬁrst part holds with K1 = 1 because of Lemma 5.2. Now, assume that ρ is a C-quasi-smooth hallway whose fastest growing edge grows of degree d = 1. Obtain a collection M of hallways by cutting ρ along the trajectories of all linearly growing edges that

Dynamics of Free Group Automorphisms

41

do not cancel within ρ. If σ is a smooth element of M, then the ﬁrst part implies that the number of edges in each σi emitted by linearly growing edges is bounded by V(σ). If σ is not smooth, then in every slice σi , the number of edges emitted by linearly growing edges is bounded by V(σ) + 2CD(ρ) (It is helpful to keep Example 3.3 in mind). Lemma 5.3 yields that M contains no more than 2CD(ρ) pieces that are not smooth. Summing up, we conclude that every slice of ρ contains at 2 most V(ρ)+(2CD(ρ)) edges emitted by linearly growing edges, so that the second statement follows with K1 = 1 and K1 (C) = 4C 2 . Now, let K be the constant from Proposition 4.2, and assume inductively that the proposition holds for some d ≥ 1. We want to ﬁnd some Kd+1 such that for all hallways ρ whose fastest growing edge is of degree d + 1, we have L(ρi ) ≤ Kd+1 V(ρ). for all slices ρi . It suﬃces to prove this with the assumption that ρ is indecomposable. Then we can perform the sawtooth construction along all trajectories of edges of degree d + 1. Since ρ is indecomposable, we obtain one C-quasi-smooth piece σ that only crosses edges of degree d or lower, so that by induction, we conclude that the number of edges in σi that were emitted by linearly growing edges is bounded by Kd V(σ) + Kd (C)Dd+1 (ρ). We conclude that L(ρi ) ≤ ≤

KV(ρ) + Kd V(σ) + Kd (C)Dd+1 (ρ) (K + Kd )V(ρ) + (2C + Kd (C)) Dd+1 (ρ).

Using Lemma 5.4, we can ﬁnd some constant M such that M V(ρ) ≥ (2C + Kd (C)) Dd+1 (ρ) for all indecomposable hallways ρ involving edges of degree d + 1. We conclude that the ﬁrst statement of the proposition holds with Kd+1 = K + Kd + M . We now prove the second assertion. Let ρ be a C-quasi-smooth hallway. We obtain two collections M1 , M2 of hallways by performing cutting and sawtooth operations as in Lemma 5.3. The elements of M1 are smooth hallways, so that for any σ ∈ M1 , the previous step yields L(σi ) ≤ Kd+1 V(σ). If σ is an element of M2 , then it is a 2C-quasi-smooth hallway, and induction yields that in every slice of σ, the number of edges emitted by linearly growing edges is bounded by Kd V(σ) + Kd (2C)D d+1 (σ).

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Summing over all elements of M1 and M2 , we conclude that every slice ρi of ρ contains at most Kd+1 V(M1 ) + Kd V(M2 ) + 2CD(ρ) · Kd (2C)Dd+1 (ρ) ≤ Kd+1 V(ρ) + 4C 2 Kd D2 (ρ) + 2CKd (2C)Dd+2 (ρ) edges emitted by linearly growing edges, so that the second statement of the proposition holds with Kd+1 (C) = 4C 2 Kd + 2CKd (2C).

Remark 5.6. The estimates of Proposition 5.5 are rather crude; lots of edges are counted several times rather than just once. I opted to present the most straightforward estimates rather than tightest ones.

6. Proof of the main result We now extend the techniques and results of Proposition 5 to arbitrary automorphisms. The presence of exponentially growing strata will turn out to be a mixed blessing. On the one hand, they make for rather simple counting arguments as polynomial contributions as in Proposition 5.5 are easily dwarfed by exponential growth. On the other hand, we will need to consider more complicated decompositions of hallways. As usual, let f : G → G be an improved relative train track map. Any statements regarding the computability of constants assume that we are given such a map. After permuting the strata as necessary, we may assume that if Hr and Hs are truly polynomial strata and r > s, then the degree of Hr is at least as large as that of Hs . Throughout this section, let K be the constant from Proposition 4.2. If Hr is an exponentially growing stratum, then we ﬁx some L > Cr , and we replace f by f M , where M is the exponent from Lemma 2.2 for this choice of L. After replacing f by a power yet again if necessary, we may assume that the image of each edge in Hr contains at least L edges in Hr . If Hr supports a closed Nielsen path τ , then the initial and terminal edges of τ are partial edges in Hr , and we may assume that the image of each of them also contains at least L edges in Hr . We say that a legal path of height r is long if it contains at least L edges in Hr . We ﬁrst record an exponential version of Lemma 5.4. Lemma 6.1. Let Hr be an exponentially growing stratum or a fast polynomial stratum. Then there exists a computable constant λ > 1 such that if σ is a circuit in Gr or a path starting and ending at ﬁxed vertices, then either σ is a concatenation of Nielsen paths of height r and subpaths in Gr−1 , or we have k (σ)) ≥ λk L(σ) + L(f#

for all k ≥ 0.

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Proof. If Hr is an exponentially growing stratum, we need to distinguish two cases: i First, assume that for some i ≥ 0, f# (σ) is a concatenation of Nielsen paths and subpaths in Gr−1 . Since σ starts and ends a ﬁxed vertices, we conclude that σ itself is a concatenation of Nielsen paths and subpaths in Gr−1 , so that there is nothing to show in this case. Let λ− , N0 be the constants from Lemma 2.5, and assume that for all i ≥ 0, i f# (σ) is not a concatenation of Nielsen paths and subpaths in Gr−1 . Let i0 be i0 (σ) contains a long legal segment. Then, using the smallest index for which f# k Lemma 2.5 and Lemma 2.2, we see that L(σ) ≥ λi−0 . Moreover, we have L(f# (σ)) ≥ k−i0 λr . 0 If we let λ = min{λ− , λr }, then we have λi−0 + λk−i ≥ λk . Hence, we have r i k 0 (σ)) ≥ λ−0 + λk−i ≥ λk . L(σ) + L(f# r If Hr = {Er } is a fast polynomial stratum, then we argue similarly, using Lemma 2.4 and Theorem 1.4, Part 4. If Hr is an exponentially growing stratum, we let Tr equal the length of the longest path in f (Hr ) ∩ Gr−1 . We ﬁx another constant Sr > 0 with the following property: Let γ be a path in Gr−1 . If L(γ) ≥ Sr , then L(f# (γ)) > 3Tr and 2 2 L(f# (γ)) > 3Tr , and if L(γ) ≤ Tr , then L(f# (γ)) < Sr and L(f# (γ)) < Sr . We can easily compute a suitable value Sr given the train track map f . We say that a path γ in Gr−1 is r-signiﬁcant if L(γ) ≥ Sr . If Hr is an exponentially growing stratum, and ρ is a C-quasi-smooth hallway of height r, then we need to develop an understanding of the lengths of components of ρi ∩Gr−1 , i.e., we need to study subpaths in Gr−1 . Intuitively, we will accomplish this by carving out subhallways in Gr−1 . Consider a maximal subpath γ ⊂ Gr−1 of some slice ρa , i.e., ρa can be expressed as αγβ, and α (resp. β) is either trivial or ends (resp. starts) with a (possibly partial) edge in Hr . We begin the construction of a new hallway ρ by letting ρ0 = γ. Now, assume inductively that we have deﬁned the slice ρi−1 such that ρi−1 is a maximal subpath of ρa+i−1 in Gr−1 (we write ρa+i−1 = αρi−1 β), and recall that the slice ρa+i is obtained by tightening μa+i f (αρi−1 β)νa+i . We deﬁne the notch μi by taking the maximal terminal subpath in Gr−1 of the path obtained from μa+i f (α) by tightening. Similarly, we deﬁne the notch ν1 by tightening the maximal initial subpath in Gr−1 of the path obtained from f (β)νa+i by tightening. Observe that tightening μi ρi−1 νi yields a maximal subpath in Gr−1 of ρa+i , and that the length of μi and νi is bounded by C + Tr . We iterate this procedure until we reach a point where tightening μi+1 f (ρi )νi+1 yields a trivial path. By applying this construction wherever possible, we obtain a fan of C + Tr quasi-smooth hallways in Gr−1 . Let M be the set of maximal elements of this fan. We let M1 be the collection of smooth hallways in M, and we let M2 be the collection of hallways in M that are not smooth.

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Let σ be an element of M2 , and assume that there exists some 0 < i < D(σ) such that L(σi ) < Sr . Then we obtain two new hallways σ , σ from σ by letting σj = σj for 0 ≤ j ≤ i and σj = σi+j for 0 ≤ j ≤ D(σ) − i; we may think of this operation as cutting σ along σi . We obtain a collection of hallways M2 by performing all possible cuts of this kind on all elements of M2 . If σ ∈ M1 ∪ M2 , we say that σ intersects a slice ρi if one of the slices of σ is a subpath of ρi . When looking for bounds on the lengths of a slice ρi , we need to ﬁnd bounds on the lengths of slices of hallways σ that intersect ρi . Deﬁnition 6.2. Fix some stratum Hr . We say that the map f satisﬁes Condition Ar if for any C ≥ 0, there exist computable constants Kr , Kr (C), and an exponent d ≥ 1, such that the following two conditions hold: • If ρ is a smooth hallway in Gr such that the slice ρ0 starts and ends at ﬁxed vertices, then L(ρi ) ≤ Kr V(ρ) for all slices ρi . • If ρ is a C-quasi-smooth hallway in Gr , then L(ρi ) ≤ Kr V(ρ) + Kr (C)Dd (ρ). If Hr is an exponentially growing stratum, then a hallway of height r is admissible if all its slices start and end at ﬁxed vertices or at points in Hr . Lemma 6.3. Let Hr be an exponentially growing stratum, and assume that Condition Ar−1 holds. Then, given some C ≥ 0, there exist computable constants C1 , C2 ≥ 1 with the following property: If ρ is an admissible C-quasi-smooth hallway of height r, then Dd (σ) L(ρi ) ≤ C1 V(ρ) + C2 σ∈M2 σ intersects ρi in an r-signiﬁcant segment

for every slice ρi of ρ. Proof. Since ρ is admissible, all slices of σ ∈ M1 start and end at ﬁxed vertices unless σ0 is contain in a zero stratum, in which case all slices σi for i > 0 start and end at ﬁxed vertices. Moreover, if σ0 is contained in a zero stratum, then L(σ1 ) = L(σ0 ). By Condition Ar−1 , we have L(σi ) ≤ Kr−1 V(σ) for all slices σi of σ ∈ M1 .

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Fix some slice ρi of ρ. Using Proposition 4.2 and Condition Ar−1 , we see that Kr−1 V(σ) L(ρi ) ≤ KV(ρ) + σ∈M1 σ intersects ρi

+

(C + Tr )Dd (σ) . Kr−1 V(σ) + Kr−1

σ∈M2 σ intersects ρi

Consider some σ ∈ M1 that intersects ρi . If the initial slice of σ is not visible in ρ, then, as we noted before, its length is bounded by Tr . Similarly, if the terminal slice of σ is not visible in ρ, then its length is also bounded by Tr . The number of elements of M1 that intersect ρi is bounded by KV(ρ). Putting it all together, we conclude that V(σ) ≤ (2KTr + 1)V(ρ). σ∈M1 σ intersects ρi

Similarly, using the fact that elements of M2 are C + Tr -quasi-smooth, and that their initial and terminal slices are either visible in ρ or of length less than Sr , we see that V(σ) ≤ (2KSr + 1)V(ρ) + 2(C + Tr ) D(σ). σ∈M2 σ intersects ρi

σ∈M2 σ intersects ρi

Since ρi contains at most KV(ρ) subpaths in Gr−1 , the total contribution of subpaths in Gr−1 that are not r-signiﬁcant is bounded by KSr V(ρ). Letting C1 = K + 2Kr−1 (K(Sr + Tr ) + 1) + KSr and C2 = Kr−1 (C + Tr ) + 2(C + Tr ), we conclude that L(ρi ) ≤ C1 V(ρ) + C2 Dd (σ). σ∈M2 σ intersects ρi in an r-signiﬁcant segment

Lemma 6.3 shows that from now on, we may focus on the polynomial contribution of nonsmooth hallways in Gr−1 that intersect a given slice ρi in an r-signiﬁcant subpath. In particular, if the initial slice ρ0 happens to be an r-legal path, then L(ρi ) ≤ C1 V(ρ) for all slices ρi since M2 is empty in this case. Lemma 6.4. Let Hr be an exponentially growing stratum, and assume that Condition Ar−1 holds. Given some C > 0, there exist computable constants C1 , C2 with the following property: If ρ is an admissible C-quasi-smooth hallway of height r, such that for every slice ρi except possibly the last one, f# (ρi ) does not contain a legal segment of length at least L, then L(ρi ) ≤ C1 V(ρ) + C2 Dd+1 (ρ) for all slices ρi .

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Proof. By Lemma 6.3, we may restrict our attention to elements of M2 that intersect a given slice ρi in an r-signiﬁcant subpath. Let D= Dd (σ). σ∈M2 σ intersects ρi in an r-signiﬁcant segment

We ﬁrst claim that the number of r-signiﬁcant subpaths in Gr−1 in a slice ρi is bounded by N (ρi ). By choice of Sr , an r-signiﬁcant subpath in Gr−1 will not cancel completely when f (ρi ) is tightened to f# (ρi ). If there were two such subpaths in one legal segment of ρi , then there would be a legal segment in Hr in between. Since we assumed that L(f (E) ∩ Hr ) ≥ L for each edge in Hr , the r-length of the image of this legal segment is at least L, which means that the slice ρi+1 contains a legal segment of length at least L, contradicting our assumption. This proves the claim if Hr does not support a closed Nielsen path, as in this case, the number of legal segments in ρi equals N (ρi ). If Hr supports a closed Nielsen path, then a legal segment of ρi that is adjacent to an illegal turn contained in a Nielsen subpath of ρi cannot contain an r-signiﬁcant subpath in Gr−1 . If such a segment contained an r-signiﬁcant subpath in Gr−1 , then f# (ρi ) would contain a legal segment of r-length L because both the initial and terminal partial edge of the Nielsen path of Hr map to legal segments of r-length at least L. This implies that the number of r-signiﬁcant subpaths in Gr−1 is bounded by N (ρi ). Now, ﬁx some slice ρi . We make the worst-case assumption that every legal segment of ρ that is not adjacent to an illegal turn contained in a Nielsen subpath contains an r-signiﬁcant subpath in Gr−1 that is a slice of a hallway σ ∈ M2 of duration j ≥ i. The number of such hallways whose duration is a given number j ≥ i is bounded by N (ρj ) + 1. We conclude that D(ρ)

D≤

N (ρj )j d .

j=i

Choosing λ according to Lemma 2.5, we conclude that N (ρi+1 ) ≤ λ−1 N (ρi )+ 1 + 2C, as ρ is C-quasi-smooth. This implies, inductively, that N (ρi ) ≤ λ−i N (ρ0 ) + 2(1 + C) We choose some B ≥

∞ j=0

i−1

λ−j ≤ λ−i N (ρ0 ) +

j=0 −j d

λ

j , and we conclude that

D(ρ)

D≤

j=0

λ (1 + 2C). λ−1

N (ρj )j d ≤ BV(ρ) +

λ (1 + 2C)Dd+1 (ρ), λ−1

since N (ρ0 ) ≤ V(ρ). If C1 , C2 are the constants from Lemma 6.3, then the lemma holds with λ (1 + 2C)C2 . C1 = C1 + C2 B and C2 = λ−1

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Deﬁnition 6.5. Let Hr be an exponentially growing stratum, and let N0 be the constant from Lemma 2.5. We say that an admissible smooth hallway ρ of height r has Property B if for all slices ρi , ρi contains no long r-legal segment, or N (ρi−1 ) < N0 . Lemma 6.6. Let Hr be an exponentially growing stratum, and assume that Condition Ar−1 holds. Let N0 be the constant from Lemma 2.5. There exist computable constants C1 , C2 with the following property: If ρ is an admissible smooth hallway of height r that satisﬁes Property B, then L(ρi ) ≤ C1 V(ρ) + C2 Dd+1 (ρ) for all slices ρi . Proof. If no slice of ρ contains a long legal segment, then the claim follows from Lemma 6.4. Otherwise, let i0 be the smallest index for which ρi0 contains a long legal segment. By choice of i0 , ρi0 −1 does not contain a long legal segment, and by hypothesis, we have N (ρi0 −1 ) < N0 . If i < i0 , then, choosing D as in the proof of Lemma 6.4, we conclude that

i 0 −1 N (ρj )j d + N0 Dd (ρ) D≤ j=0

≤ BV(ρ) + N0 Dd+1 (ρ), so that the lemma holds for all ρi with i < i0 . For i ≥ i0 , ρi splits as a concatenation of long r-legal paths and subpaths that contain illegal turns and no long legal subpaths. Each slice may, conceivably, contain slices of N (ρi0 −1 ) < N0 hallways of duration D(ρ). The polynomial contribution of these hallways is bounded by N0 Dd (ρ). In addition, the number of short legal segments around illegal turns is at most 2N0 . Each of them contains not more than one r-signiﬁcant subpath in Gr−1 , belonging to a hallway of duration at most D(ρ) − i0 . The polynomial contribution of these paths is bounded by 2N0 (D(ρ) − i0 )d . Now, since ρi0 contains a long legal segment, the length of D(ρ)−i0

ρD(ρ) = f#

(ρi0 )

D(ρ)−i

0 is at least λr . We can easily ﬁnd some B > 0 such that B λkr ≥ 2N0 k d for all k ≥ 0. We conclude that for the sum of all polynomial contributions in ρi , we have 2N0 (D(ρ) − i0 )d + N0 D d (ρ) ≤ B V(ρ) + N0 Dd (ρ),

which completes the proof of the lemma.

The remaining two lemmas deal with arbitrary smooth hallways of height r as well as quasi-smooth hallways by essentially decomposing them into pieces of the kind that we analyzed in the previous lemmas.

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Lemma 6.7. Let Hr be an exponentially growing stratum, and assume that Condition Ar−1 holds. Then there exist computable constants C1 , C2 with the following property: If ρ is a smooth admissible hallway of height r, then L(ρi ) ≤ C1 V(ρ) + C2 Dd+1 (ρ) for all slices ρi . Proof. Let λ− , N0 be the constants from Lemma 2.5. As in the proof of Lemma 6.1, ≥ we let λ = min{λ− , λr }, and we remark that for 0 ≤ j ≤ k, we have λj− +λk−j r λk . This basic estimate will be crucial in the proof of this lemma. We choose some B > 0 such that Bλk > k d+1 for all k ≥ 0. Let C1 , C2 be the maximum of the corresponding constants from the previous lemmas. We will see that the lemma holds with C1 = C1 + 3BC2 and C2 = C2 . We ﬁrst observe that if ρ satisﬁes Property B, then the lemma follows from Lemma 6.6. If ρ0 contains long legal segments, we can split ρ0 into long r-legal subpaths and neighborhoods of illegal turns (i.e., illegal turns surrounded by legal paths whose length is at most C2r ). Split ρ0 as ρ0 = α0;1 β0;1 α0;2 · · · α0;m β0;m , where all subpaths α0;i are long legal segments, and all subpaths β0;i are neighborhoods of illegal turns. Such a decomposition of ρ0 induces a decomposition of ρ into hallways, and we can choose the decomposition such that all resulting pieces are admissible, and that the legal segments are as long as possible, subject j j to admissibility. We write αj;i = f# (α0;i ) and βj;i = f# (β0;i ). Let k = D(ρ). For each long legal subpath α0;i , Lemma 6.3 yields that L(αj;i ) ≤ C1 (L(α0;i ) + L(αk;i )), for all 0 ≤ j ≤ k. Since α0;i is a long legal segment, we have L(αk;i ) ≥ λkr ≥ λk . If the hallway deﬁned by β0;i satisﬁes Property B, then we have L(βj;i ) ≤ C1 (L(β0;i ) + L(βk;i )) + C2 k d+1 , and we have k d+1 ≤ BL(αk;i ), hence L(βj;i ) ≤ C1 (L(β0;i ) + L(βk;i )) + BC2 L(αk;i ), i.e., we can ﬁnd a legal segment adjacent to β0;i whose contribution to the visible edges of ρ dominates the possible polynomial contribution of β0;i . This takes care of the long legal segments in ρ0 as well as the subpaths that satisfy Property B. Hence, we only need to deal with those paths that do not satisfy Property B. Assume that for some 0 ≤ i ≤ m, β0;i is one of them. Then there exists some j0 such that βj0 ;i contains a long legal segment, but βj0 −1;i does not, and N (βj0 −1;i ) ≥ N0 . As before, we split βj0 ;i into long legal segments and neighborhoods of illegal turns, obtaining a decomposition βj0 ,i = αj0 ;i,0 βj0 ;i,0 · · · αj0 ;i,m βj0 ;i,m , where αj0 ;i,k are r-legal subpaths, and βj0 ;i,k are neighborhoods of illegal turns. We can ﬁnd splittings βj;i = αj;i,0 βj;i,0 · · · αj;i,m βj;i,m for all 0 ≤ j ≤ k, such that f# (αj;i,k ) = αj+1;i,k and f# (βj;i,k ) = βj+1;i,k . We may choose those splitting such that the resulting pieces are admissible, and such that the legal segments αj0 ;i,k are as long as possible, subject to admissibility.

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Now, ﬁx on one subpath αj0 ;i,k . If N is the number of r-signiﬁcant subpaths in Gr−1 in αj0 ;i,k , then αj0 −1;i,k contains at least N legal segments containing r-signiﬁcant subpaths in Gr−1 . By Lemma 2.5, we have L(β0;i ) ≥ N (βj0 ;i ) ≥ λj−0 −1 N (βj0 −1;i ), so that we can ﬁnd λj−0 −1 N illegal turns in β0;i , and we can ﬁnd j0 −1 0 0 λk−j edges in βk,i . Using our earlier estimate, we see that (λ− + λk−j )N ≥ r r −1 k λ− λ N . The polynomial contribution of the r-signiﬁcant subpaths in Gr−1 of αj0 ;i,k is bounded by Kr−1 (Tr )N k d+1 ≤ BKr−1 (Tr )N λk , i.e., it is dominated by corresponding visible edges. This leaves us to deal with the adjacent subpaths βj0 ;i,k and βj0 ;i,k−1 . If β0;i,k satisﬁes Property B, then its polynomial contribution is bounded by C2 k d+1 , which in turn is bounded by BC2 λk . This takes care of the legal segments αj0 ;i,k as well as those neighborhoods of illegal turns that satisfy Property B. We apply the previous reasoning to the remaining paths βj0 ;i,k , completing the proof of the lemma. Lemma 6.8. Let Hr be an exponentially growing stratum, and assume that Condition Ar−1 holds. Given some C > 0, there exist computable constants C1 , C2 with the following property: If ρ is an admissible C-quasi-smooth hallway of height r, then L(ρi ) ≤ C1 V(ρ) + C2 Dd+3 (ρ) for all slices ρi . Proof. The idea of this proof is to decompose the hallway ρ into pieces that are either smooth or C-quasi-smooth satisfying the hypothesis of Lemma 6.4. In order to ﬁnd this decomposition, we introduce trajectories of points in Hr . This deﬁnition may be aﬀected by the choices made when tightening (Remark 1.1). In order to avoid ambiguities, for each index 1 ≤ i < D(ρ), we ﬁx a sequence of elementary cancellations that turn μi ρi−1 νi into ρi . If p is a point in ρi ∩ Hr , we consider its image f (p) in f (ρi ). We say that p survives if f (p) is contained in Hr and if f (p) is not contained in an edge that cancels when f (ρi ) is tightened to f# (ρi ). If p survives, then f (p) is contained in ρi+1 , or it is contained in the parts of f# (ρi ) that cancel when μi+1 f# (ρi )νi+1 is tightened to ρi+1 . Thinking of the hallway ρ as spanning a (possibly singular) disk, we draw a line segment (in this disk) from the surviving points in each slice to their images. If p is a point in a visible edge such that p and all its images survive, then p deﬁnes a line starting and ending in visible edges, called the trajectory of p. The trajectories of two points need not be disjoint, but that does not concern us here. We say that two trajectories are parallel if their initial points are both contained in ρ0 or both contained in the same notch, and if their terminal points are both contained in ρD(ρ) or both contained in the same notch. The crucial observation is that equivalence classes of parallel trajectories are closed subsets of the disk spanned by ρ, so that in every equivalence class, we can ﬁnd trajectories of

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two points p1 , p2 that are extremal in the following sense: If p is a point whose trajectory is parallel to those of p1 and p2 , then p is located between p1 and p2 . We now cut ρ along the extremal trajectories of all equivalence classes of parallel trajectories, obtaining pieces that are either smooth or C-quasi-smooth. Moreover, all the resulting pieces are admissible. Let M1 be the collection of smooth pieces and M2 the collection of pieces that are not smooth. Note that V(M1 ) + V(M2 ) = V(ρ). We now claim that all elements of M2 satisfy the hypothesis of Lemma 6.8. Suppose otherwise, i.e., there exists some σ ∈ M2 such that for some slice σi , f# (σi ) contains a legal segment of length at least L. Within the interior of this legal segment, we can ﬁnd some point p such that all images of p survive in subsequent slices. Since p is the image of surviving points, we obtain a trajectory along which we can cut σ, contradicting the fact that we obtained σ by cutting ρ along extremal trajectories. By Lemma 6.7, there are constants C1 , C2 such that for every σ ∈ M1 and every slice σi of σ, we have L(σi ) ≤ C1 V(σ) + C2 Dd+1 (σ), and by Lemma 6.8, there are constants C1 , C2 such that L(σi ) ≤ C1 V(σ) + C2 Dd+1 (σ) for every slice σi of every σ ∈ M2 . There are at most 2(D(ρ) − 1) notches, so that the number of equivalence classes of parallel trajectories is bounded by (2(D(ρ) − 1) + 1)2 (another extremely crude estimate, but it’ll do). Since we cut along no more than two trajectories per equivalence class, we obtain no more than 2 (2 (D(ρ) − 1) + 1)2 + 1 ≤ 8D2 (ρ) pieces. Letting C1 = max{C1 , C1 } and C2 = 8 max{C1 , C2 }, we conclude that L(ρi ) ≤ C1 V(ρ) + C2 Dd+3 (ρ) for all slices of ρ.

We now have all the ingredients that we need to prove Theorem 1.9. Proof of Theorem 1.9. We ﬁrst show that Condition Ar holds for all strata Hr . This implies, in particular, that the second statement of Theorem 1.9 holds for paths starting and ending at ﬁxed vertices. If ρ is a path starting and ending at arbitrary vertices, then Theorem 1.4, Part 2 yields that f# (ρ) starts and ends at ﬁxed vertices, so that, in fact, the second statement of Theorem 1.9 follows from Condition Ar in this case as well. We note that Condition A0 holds trivially, and we assume inductively that Condition Ar−1 holds for some r. We want to prove Condition Ar . Assume that Hr is an exponentially growing stratum, and let ρ be a smooth hallway of height r such that ρ0 starts and ends at ﬁxed vertices. If ρ0 is a concatenation of Nielsen paths of height r and paths in Gr−1 , then we can split ρ0 at

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the endpoints of its subpaths in Gr−1 , and Condition Ar−1 completes the proof. We now assume that ρ0 is not a concatenation of Nielsen paths and paths in Gr−1 . By Lemma 6.7, we have constants C1 , C2 such that L(ρi ) ≤ C1 V(ρ) + C2 Dd+1 (ρ) for all slices ρi . Moreover, by Lemma 6.1, there exists some C > 0 and λ > 1, independently of ρ, such that V(ρ) ≥ CλD(ρ) . We can easily ﬁnd some constant B such that BCλk ≥ C2 k d+1 for all k ≥ 0. Now the ﬁrst part of Condition Ar follows, with Kr = C1 + B. Lemma 6.8 yields the second part of Condition Ar , so that Condition Ar holds. We now assume that Hr is a polynomially growing stratum. Because of Proposition 5.5, we only need to consider the following situation: Either Hr is fast, or Hr is truly polynomial, but ρ contains fast polynomial edges or non-Nielsen subpaths in exponentially growing strata. In order to see that the second part of Condition Ar holds for a C-quasismooth hallway ρ of height r, we apply cutting and sawtooth constructions to ρ, obtaining a collection of (C + |ur |)-quasismooth hallways of height r − 1 or less, so that the second part of Condition Ar immediately follows from the second part of Condition Ar−1 . Now, given a smooth hallway ρ of height r, we apply cutting and sawtooth constructions again, obtaining a collection of 2|ur |-quasismooth hallways. For each slice ρi , the second part of Condition Ar−1 yields a polynomial bound on the number of edges marked by linear strata (Deﬁnition 4.1). Now, since either Hr is fast or ρ contains fast polynomial edges or non-Nielsen subpaths in exponentially growing strata, Lemma 6.1 provides an exponential lower bound for the number of visible edges. As before, the exponential lower bound for visible edges easily dominates the polynomial lower bound for edges marked by linear strata, which completes the proof of Condition Ar . Finally, in order to prove the ﬁrst part of Theorem 1.9, we need to understand the dynamics of circuits. Let σ be a circuit of height r. If Hr is a polynomially growing stratum, then Remark 1.6 yields that σ splits, at ﬁxed vertices, into basic paths of height r and paths in Gr−1 , so that Condition Ar proves the claim. Assume that Hr is an exponentially growing stratum. If σ is a concatenation of Nielsen paths of height r and paths in Gr−1 , then we can split σ at the endpoints of its subpaths in Gr−1 , so that Condition Ar−1 completes the proof in this case. We now assume that σ is not a concatenation of Nielsen paths and subpaths in Gr−1 . Then σ splits at a point p in Hr , so that we may interpret σ as a path starting and ending at v. Let ρ be a smooth hallway with ρ0 = σ. Then, by Lemma 6.7, we can ﬁnd constants C1 , C2 such that L(ρi ) ≤ C1 V(ρ) + C2 Dd (ρ)

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for all slices ρi . Moreover, by Lemma 6.1, we can ﬁnd constants C, λ such that V(ρ) ≥ CλD(ρ) . As before, we ﬁnd some constant B such that BCλk ≥ C2 k d for all k ≥ 0, so that the ﬁrst statement of Theorem 1.9 holds with Kr = C1 + B. Finally, if ρ0 is a Nielsen path of height r, then there is nothing to show. This completes the proof.

Glossary For the convenience of the reader, we brieﬂy list some of the more technical notions used in this paper. L(ρ) length of a path ρ in a conveniently chosen metric (Remark 1.8) λr growth rate of an exponentially growing stratum Hr Cr critical length of an exponentially growing stratum Hr (Equation 2) Rr eigenray in a polynomially growing stratum Er (Equation 1) n(ρ) number of legal segments in a path ρ in an exponentially growing stratum N (ρ) number of legal segments that do not overlap with closed Nielsen subpath of ρ M exponent from Lemma 2.2 V(w) visible length (Deﬁnition 3.8) of a hallway (Deﬁnition 3.1) D(w) duration of a hallway Condition Ar technical condition (Deﬁnition 6.2) Property B technical condition (Deﬁnition 6.5)

References [BF92]

M. Bestvina and M. Feighn. A combination theorem for negatively curved groups. J. Diﬀerential Geom., 35(1):85–101, 1992. [BFH97] M. Bestvina, M. Feighn, and M. Handel. Laminations, trees, and irreducible automorphisms of free groups. Geom. Funct. Anal., 7(2):215–244, 1997. [BFH00] Mladen Bestvina, Mark Feighn, and Michael Handel. The Tits alternative for Out(Fn ). I. Dynamics of exponentially-growing automorphisms. Ann. of Math. (2), 151(2):517–623, 2000. [BG] Martin R. Bridson and Daniel P. Groves. The quadratic isoperimetric inequality for mapping tori of free group automorphisms. arXiv:0802.1323. [BH92] Mladen Bestvina and Michael Handel. Train tracks and automorphisms of free groups. Ann. of Math. (2), 135(1):1–51, 1992. [Bri] Peter Brinkmann. Detecting automorphic orbits in free groups. arXiv:0806.2889v1.

Dynamics of Free Group Automorphisms [Bri00] [Coo87] [DV96]

[Gan59] [Ger94] [Mac00]

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Peter Brinkmann. Hyperbolic automorphisms of free groups. Geom. Funct. Anal., 10(5):1071–1089, 2000. arXiv:math.GR/9906008. Daryl Cooper. Automorphisms of free groups have ﬁnitely generated ﬁxed point sets. J. Algebra, 111(2):453–456, 1987. Warren Dicks and Enric Ventura. The group ﬁxed by a family of injective endomorphisms of a free group. American Mathematical Society, Providence, RI, 1996. F.R. Gantmacher. The theory of matrices. Vols. 1, 2. Chelsea Publishing Co., New York, 1959. Translated by K.A. Hirsch. S.M. Gersten. The automorphism group of a free group is not a CAT(0) group. Proc. Amer. Math. Soc., 121(4):999–1002, 1994. N. Macura. Quadratic isoperimetric inequality for mapping tori of polynomially growing automorphisms of free groups. Geom. Funct. Anal., 10(4):874–901, 2000.

Peter Brinkmann Department of Mathematics The City College of CUNY New York, NY 10031, USA e-mail: [email protected]

Combinatorial and Geometric Group Theory Trends in Mathematics, 55–91 c 2010 Springer Basel AG

Geodesic Rewriting Systems and Pregroups Volker Diekert, Andrew J. Duncan and Alexei G. Myasnikov Abstract. In this paper we study rewriting systems for groups and monoids, focusing on situations where ﬁnite convergent systems may be diﬃcult to ﬁnd or do not exist. We consider systems which have no length increasing rules and are conﬂuent and then systems in which the length reducing rules lead to geodesics. Combining these properties we arrive at our main object of study which we call geodesically perfect rewriting systems. We show that these are well behaved and convenient to use, and give several examples of classes of groups for which they can be constructed from natural presentations. We describe a Knuth-Bendix completion process to construct such systems, show how they may be found with the help of Stallings’ pregroups and conversely may be used to construct such pregroups. Mathematics Subject Classiﬁcation (2000). 68Q42, 20F05, 20M32, 20E06. Keywords. String rewriting systems, Geodesically Perfect, Knuth-Bendix, Stallings pregroups.

1. Introduction A presentation of a group or monoid may be thought of as a rewriting system which, in certain cases, may give rise to algorithms for solving classical algorithmic problems. For example if the rewriting system is ﬁnite and convergent (that is, conﬂuent and terminating) then it can be used to solve the word problem and to ﬁnd normal forms for elements of the group. This is one reason for the importance of convergent rewriting systems in group theory. However there are many well behaved, algorithmically tractable, groups for which natural presentations do not give rise to convergent rewriting systems. In this paper we investigate properties of rewriting systems, which are not in general ﬁnite or terminating, but which all the same give algorithms for such tasks as solving the word problem, computation of normal forms or computation of geodesic representatives of group elements. We Part of this work was begun in 2007 when the ﬁrst and third author were at the CRM (Centro Recherche Matem` atica, Barcelona) at the invitation of Enric Ventura.

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contend that the resulting algorithms are often more convenient and practical than those arising from more conventional ﬁnite convergent systems. Rewriting methods in algebra have a very long and rich history. In groups and semigroups they are usually related to the word problem and take their roots in the ground breaking works of Dehn and Thue (not to mention the classical Euclidean and Gaussian elimination algorithms!). Several famous algorithms in group theory are in fact particular types of string rewriting processes: the Nielsen method in free groups, Hall collection in nilpotent and polycyclic groups, the Dehn algorithm in small cancellation and hyperbolic groups, Tits rewriting in Coxeter groups, convergent rewriting systems for ﬁnite groups, and so on. In rings and algebras rewriting methods appear as a major tool in computing normal forms of elements [40, 45, 10], for solving the word and ideal-membership problems. These techniques emerged independently in various branches of algebra at diﬀerent times and under diﬀerent names (the diamond lemma, Gr¨ obner or Shirshov bases, Buchberger’s algorithm and S-polynomials, for instance). They have gained prominence with the progress of practical computing, as real applications have become available. Notably, crucial developments in methods of computational algebra originated in commutative algebra and algebraic geometry, with Buchberger’s celebrated algorithm and related computational techniques, which revolutionised the whole area of applications. We refer to [11], and the references therein, for more details. From the theoretical view point the main shift in the paradigm came with the seminal paper of Knuth and Bendix [33]. In this paper they introduced a process, now known as the Knuth-Bendix (KB) procedure, which uniﬁed the ﬁeld of rewriting techniques in (universal) algebra. The KB procedure gives a solid theoretical basis for practical implementations, even though the procedure itself may lead to non-optimal algorithms for solving word problems. Roughly speaking a KB procedure takes as input a ﬁnite system of identities (between terms) and a computable (term) ordering such that the identities can be read as a ﬁnite set of directed rewrite rules. Using the crucial concept of critical pairs the procedure adds in each round more and more rules, and it stops only if the system is completed. Thus the KB procedure attempts to construct an equivalent convergent (term) rewriting system: which in particular allows unique normal forms to be found by a simple strategy. When the procedure terminates we obtain a solvable word problem. In the case of commutative algebra this concept can be viewed as Buchberger’s algorithm and termination is guaranteed. In case of algebraic structures like groups or monoids we have a special case of a term rewriting system since the rewriting process is based on strings. (Formally, monoid generators are read as unary function symbols, and the neutral element is read as a constant.) As has been mentioned above, the history of rewriting systems in monoids and groups is about one hundred years old, with the main focus on convergent rewriting systems and algorithms for computing normal forms. Any presentation M = Γ | i = ri (i ∈ I) of a monoid M gives a rewriting system S = {i → ri ,i ∈ I}

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which deﬁnes M via the congruence relation it generates on the free monoid Γ∗ . Every rule → r ∈ S allows one to rewrite a word uv into the word urv and this gives a (non-deterministic) word rewriting procedure associated with S. If the system S is convergent (see Section 2) then this rewriting system describes a deterministic algorithm which computes the normal forms of elements, thus solving the word problem in the monoid M . This yields the major interest in ﬁnite convergent systems. Many groups are known to allow ﬁnite convergent systems (for example Coxeter groups, polycyclic groups, some small cancellation groups: see books [30, 46, 34] for more examples and details). The primary task here is to ﬁnd a ﬁnite convergent system for a given ﬁnitely presented monoid, assuming that such a system exists. In principle the KB procedure performs this task. However, several obstacles may present themselves. By design, to start the KB procedure one has to ﬁx in advance an ordering on Γ∗ , with particular properties, as described in Section 2.3. This may seem like a minor hurdle but the diﬃculty is that, even for well-understood groups with two orderings which look very much alike, it may happen that using the ﬁrst the KB process halts and outputs a convergent system while with respect to the second there exists no ﬁnite convergent system: see Example 2.5 below. Furthermore, the existence of a ﬁnite convergent system also depends on a choice of the set of generators of the group. This means that for KB to succeed one has to make the right choice of a set of generators Γ and of an ordering on Γ∗ . In fact [42] in general the problem of whether or not a given ﬁnitely presented group can be deﬁned by a ﬁnite convergent rewriting system is undecidable. In addition, even when restricted to instances where the generators and the order have been chosen so that the KB process will halt giving a ﬁnite convergent rewriting system, there may be no eﬀectively computable upper bound on the running time of the KB procedure. To make things even more interesting, having a ﬁnite convergent rewriting system S does not guarantee a fast solution of the word problem in the monoid M (see Section 2.3). All these results show that the KB process for ﬁnite convergent systems, while being an important theoretical tool, is not a panacea for problems in computational algebra. As a ﬁrst step towards resolving some of these diﬃculties we consider, in Section 4, the class of preperfect rewriting systems: that is those which are conﬂuent and have no length increasing rules. These restrictions are enough to allow solution of the word problem and to ﬁnd geodesic representatives and, as examples show, such systems are common in geometric group theory. In fact in Section 7 we describe preperfect rewriting systems for Coxeter groups, graph groups, HNN-extensions and free products with amalgamation. One disadvantage of these systems is that it is undecidable whether a ﬁnite rewriting system is preperfect or not [38] (see Theorem 4.6). Another desirable property of rewriting systems is that they should be geodesic; meaning that shortest representatives of elements can be found by applying only the length reducing rules of the system. A group deﬁned by a ﬁnite geodesic rewriting system has solvable word problem and in [24] these groups are characterised as the ﬁnitely generated virtually free groups. However, as we show in

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Section 5.1, the question of a whether or not a ﬁnite rewriting system is geodesic is undecidable. Combining properties of preperfect and geodesic rewriting systems we arrive at geodesically perfect rewriting systems (deﬁned in Section 5.2). These were ﬁrst investigated by Nivat and Benois [41] where they were called quasi-parfaites. Elsewhere these rewriting systems are also known as almost conﬂuent, see for example [6] but here we prefer the notation geodesically perfect since these systems are designed to deal with geodesics in groups and monoids. In [41] it was shown that the property of being geodesically perfect is decidable for ﬁnite systems. This leads to a new Knuth-Bendix completion procedure for constructing geodesically perfect systems as we explain below. One advantage of this KB process is that it requires no choice of ordering, using only the partial order given by word length in Γ∗ . Among the examples of Section 7 are rewriting systems for amalgamated products and HNN-extensions. As several important frameworks have been developed to unify the studies of such groups (Bass-Serre Theory, pregroups and relatively hyperbolic groups, for example) it is natural to look for a uniﬁed theory of rewriting systems covering HNN-extensions and amalgamated products. In this paper, following Stallings [49, 50], we approach this uniﬁcation question from a combinatorial view-point via pregroups and their universal groups: which seem to lend themselves naturally to algorithmic and model theoretic problems. Intuitively, a pregroup can be viewed as a “partial group”, that is, a set P with a partial (not everywhere deﬁned) multiplication m : P × P → P , or a piece of the multiplication table of some group, that satisﬁes some particular axioms. In this case the universal group U (P ) can be described as the group deﬁned by the presentation with a generating set P and a set of relations xy = z for all x, y ∈ P such that m(x, y) is deﬁned and equal to z. On the other hand, Stallings proved that U (P ) can be realized constructively as the set of all P -reduced forms (reduced sequences of elements of P ) modulo a suitable equivalence relation and a naturally deﬁned multiplication. We discuss these deﬁnitions in detail in Section 8. In Section 8.1 we show how the existence of a pregroup allows us to construct a preperfect rewriting system for the universal group. Moreover, we show in Theorem 8.4 that this system is geodesically perfect. In this way pregroups may play a role in clarifying completion procedures of KB type. In particular, completing a given presentation (in terms of generators and relators) of a group G to a larger presentation, which is a pregroup, amounts to a construction of a geodesically perfect rewriting system for G. As an application of these results we obtain a slight strengthening of the result of [24]. It is known that a group G is virtually free if and only if G = U (P ) for a ﬁnite pregroup P [44] and combining this result with Theorem 8.4 we see that a ﬁnitely generated group is virtually free if and only if it is deﬁned by a geodesically perfect rewriting system (Corollary 8.7).

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2. Rewriting techniques 2.1. Basics In this section we recall the basic concepts from string rewriting. We use rewriting techniques as a tool to prove that certain constructions have the expected properties. A rewriting relation over a set X is a binary relation =⇒⊆ X × X. We denote ∗ by =⇒ the reﬂexive and transitive closure of =⇒, by ⇐⇒ its symmetric closure ∗

and by ⇐⇒ its symmetric, reﬂexive, and transitive closure. We also write y ⇐= x ≤k

whenever x =⇒ y, and we write x =⇒ y whenever we can reach y in at most k steps from x. Deﬁnition 2.1. The relation =⇒⊆ X × X is called: ≤1

≤1

i) strongly conﬂuent, if y⇐=x=⇒z implies y =⇒ w ⇐= z for some w, ∗

∗

∗

∗

ii) conﬂuent, if y ⇐= x =⇒ z implies y =⇒ w ⇐= z for some w, ∗

∗

∗

iii) Church-Rosser, if y ⇐⇒ z implies y =⇒ w ⇐= z for some w, ∗

∗

iv) locally conﬂuent, if y⇐=x=⇒z implies y =⇒ w ⇐= z for some w. The following facts are well known and can be found in several text books (see for example [6, 31]). 1) Strong conﬂuence implies conﬂuence. 2) Conﬂuence is equivalent to Church-Rosser. 3) Conﬂuence implies local conﬂuence but the converse is false, in general. 2.2. Rewriting in monoids Rewriting systems over monoids (and in particular over groups) play an important part in algebra. Let M be a monoid. A rewriting system over M is a binary relation S ⊆ M × M . It deﬁnes the rewriting relation =⇒ ⊆ M × M such that S

x=⇒y if and only if x = pq, y = prq for some (, r) ∈ S. S

∗

∗

S

S

The relation ⇐⇒ ⊆ M × M is a congruence on M , hence the quotient set M/ ⇐⇒ forms a monoid with respect to the multiplication induced from M . We denote it by M/ { = r | (, r) ∈ S } or, simply by M/S or MS . Two rewriting systems S ∗ ∗ and T over a monoid M are termed equivalent if ⇐⇒ = ⇐⇒, i.e., MS = MT . S

T

We say that a rewriting system S is strongly conﬂuent (or conﬂuent, etc.) if the relation =⇒ has the corresponding property. Instead of (, r) ∈ S we also S

write −→r ∈ S and ←→r ∈ S in order to indicate that both (, r) and (r, ) are in S.

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We say that a word w is S-irreducible (sometimes we omit S here), if no ∗ left-hand side of S occurs in w as a factor. Thus, if w is irreducible then w =⇒ w S

implies w = w . The set of all irreducible words is denoted by IRR(S). In order to compute with monoids (in particular, groups) we usually specify a choice of monoid generators Γ, sometimes called an alphabet. For groups we often assume that Γ is closed under inversion, so Γ = Σ ∪ Σ−1 where Σ is a set of group generators. For an alphabet Γ we denote by Γ∗ the free monoid with basis Γ. Throughout, 1 denotes the neutral element in monoids or groups. In particular, 1 is also used to denote the empty word in a free monoid Γ∗ . If we can write w = xuy, then we say that u is a factor of w. For free monoids a factor is sometimes also called a subword, but this might lead to confusion because other authors understand by a subword simply a subsequence or scattered subword. Rewriting systems S over a free monoid Γ∗ are sometimes called string rewriting systems or semi-Thue systems. In this case the quotient Γ∗ /S has the standard monoid presentation Γ | { = r | (, r) ∈ S }. We say that a string rewriting system S deﬁnes a monoid M if Γ∗ /S is isomorphic to M . In addition, if P is a property of rewriting systems (Church-Rosser, strongly conﬂuent, conﬂuent, etc.) we say that a monoid M has a P -presentation if it can be deﬁned by a system with property P . For groups two types of presentations via generators and relators arise: monoid presentations, described above, and group presentations, typical in combinatorial group theory and topology. More precisely, we say that G = Γ∗ /S is a monoid presentation of a group G if the alphabet Γ is of the form Γ = Σ ∪ Σ−1 , where Σ is a set of group generators, and Σ−1 = {σ −1 | σ ∈ Σ} is the set of formal inverses of Σ (in which case Γ∗ is a the free monoid with an involution σ → σ−1 ). Given a group presentation X | R of a group G one can easily obtain a monoid presentation of G by adding the formal inverses X −1 to the set of generators X of G and the “trivial” relations xx−1 = 1, x−1 x = 1, x ∈ X} to the relators of G. We consider here monoid presentations of groups, except where explicitly indicated otherwise. 2.3. Convergent rewriting systems In this section we brieﬂy discuss convergent (or complete) rewriting systems, which play an important role in algebra due to their relation to normal forms. A relation =⇒⊆ X × X is called terminating (or Noetherian) if every inﬁnite chain ∗

∗

∗

∗

x0 =⇒ x1 =⇒ · · · xi−1 =⇒ xi =⇒ · · · becomes stationary. There are two typical sources of terminating string rewriting systems S ⊆ Γ∗ × Γ∗ . Systems of the ﬁrst type are length-reducing, i.e., for any rule → r ∈ S one has || > |r|, where |x| is the length of a word x ∈ Γ∗ . Systems of the second type are compatible with a given reduction ordering on Γ∗ , which means that if → r ∈ S then r. Recall that a reduction ordering on Γ∗ is a well-ordering

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preserving left and right multiplication (i.e., if u v then aub avb for any a, b ∈ Γ∗ ). Clearly, such systems are terminating. In fact, the condition that S is compatible with some partial order, , preserving left and right multiplication is just a reformulation of the terminating property. Indeed, if S is terminating ∗ then there is a binary relation S on Γ∗ deﬁned by u S v if and only if u =⇒ v. S

In this case S is a partial well-founded ordering (no inﬁnite descending chains), such that S r for any rule → r ∈ S. Moreover, the converse is also true. (The condition that is total is not needed here but is required in running the Knuth-Bendix completion procedure, see below). A relation =⇒⊆ X × X is called convergent (or complete) if it is locally conﬂuent and terminating. The following properties are crucial. Let S be a convergent rewriting system. 1) S is conﬂuent (see for example [6, 31]). ∗ 2) Every ⇐⇒ equivalence class in Γ∗ contains a unique S-reduced word (a word S

to which no rule from S is applicable). 3) If S is ﬁnite then for a given word w ∈ Γ∗ one can eﬀectively ﬁnd its unique S-reduced form (just by subsequently rewriting the word w until the result is S-reduced). The results above show that if a monoid M has a ﬁnite convergent presentation then the word problem in M , as well as the problem of ﬁnding the normal forms, is decidable. This explains the popularity of convergent systems in algebra. There are many examples of groups that have ﬁnite convergent presentations: ﬁnite groups, polycyclic groups, free groups and some geometric groups (see [46, 20, 34] for details). One of the major results on convergent systems concerns the Knuth-Bendix procedure (KB) (see [6] for general rewriting systems and [46, 20] for groups), which can be stated as follows. Let be a reduction well-ordering on Γ∗ and S ⊆ Γ∗ × Γ∗ a ﬁnite rewriting system compatible with . If there exists a ﬁnite convergent rewriting system T ⊆ Γ∗ × Γ∗ compatible with which is equivalent to S, then, in ﬁnitely many steps, the Knuth-Bendix procedure KB ﬁnds a ﬁnite convergent rewriting system S ⊆ Γ∗ × Γ∗ compatible with which is also equivalent to S. There are three principle remarks due here. Remark 2.2. The time complexity of the word problem in a monoid MS deﬁned by a ﬁnite convergent system S may be of an arbitrarily high complexity [43]. Remark 2.3. It may happen that the word problem in a monoid MS deﬁned by a ﬁnite convergent system S is decidable in polynomial time, whereas the complexity of the standard rewriting algorithm that ﬁnds the S-reduced forms of words can be of an arbitrarily high complexity [43]. These remarks show that convergent rewriting systems may not be the best tool to deal with complexity issues related to word problems and normal forms in monoids.

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Remark 2.4. The Knuth-Bendix procedure really depends on the chosen ordering . The following example shows that in a free Abelian group of rank two the KB procedure relative to one length-lexicographic ordering results in a ﬁnite convergent presentation, while another length-lexicographic ordering does not allow any ﬁnite convergent presentations for the same group. Example 2.5 ([21], page 127). Let G be the free Abelian group given by the following monoid presentation.

x, y, x−1 , y −1 | xy = yx, xx−1 = x−1 x = yy −1 = y −1 y = 1. Then the KB procedure with respect to the length-lexicographic ordering induced by the ordering x < x−1 < y < y −1 of the generators outputs a ﬁnite convergent system deﬁning G: xx−1 =⇒ 1, x−1 x =⇒ 1, yy −1 =⇒ 1, y −1y =⇒ 1, yx =⇒ xy, y −1 x =⇒ xy −1 , yx−1 =⇒ x−1 y, y −1 x−1 =⇒ x−1 y −1 . However, there are no ﬁnite convergent systems deﬁning G and compatible with the length-lexicographic ordering x < y < x−1 < y −1 . Therefore, even if a ﬁnite convergent presentation for a monoid M exists it ´ unmight be hard to ﬁnd it using the Knuth-Bendix procedure. In addition O’D´ laing [42] has shown that the problem of whether or not a given ﬁnitely presented group can be deﬁned by a ﬁnite convergent rewriting system is undecidable. It is not hard to see that all ﬁnitely generated commutative monoids have a ﬁnite convergent presentation, [13]. However, this demands enough generators, in general. For example, a free Abelian group of rank k can be generated as a monoid by an alphabet of size k + 1, but in order to ﬁnd a ﬁnite convergent system for it we need at least 2k generators, see [14]. Another nice example of this kind is the non-commutative semi-direct product of Z by Z. Even as a monoid we need just two generators a and b and one relation abba = 1. There is no ﬁnite convergent ∗ ∗ ∗ ∗ system S ⊆ { a, b } × { a, b } such that { a, b } / { abba = 1 } = { a, b } /S, but clearly such systems exist if we allow more generators. See [31] for more details about this example. We ﬁnish the section with a few open problems. Problem 2.6. Is it true that every hyperbolic group has a ﬁnite convergent presentation? It is known that some hyperbolic groups have ﬁnite convergent presentations, for example, surface groups [34]. Problem 2.7. Is it true that every ﬁnitely generated fully residually free group has a ﬁnite convergent presentation? The next two problems are from [43]. Problem 2.8. Do all automatic groups have ﬁnite convergent presentations?

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Problem 2.9. Do all one-relator groups have ﬁnite convergent presentations? Notice that all the groups above satisfy the homological condition F P∞ ; which is the main known condition necessary for a group to have a ﬁnite convergent presentation, see [48, 47]. 2.4. Computing with inﬁnite systems In this section we discuss computing with inﬁnite systems. An inﬁnite string rewriting system S ⊆ Γ∗ × Γ∗ can be used in computation if it satisﬁes some natural conditions. Firstly, one has to be able to recognize if a given pair (u, v) ∈ Γ∗ × Γ∗ gives a rule u → v ∈ S or not, i.e., the system S must be a recursive subset of Γ∗ × Γ∗ . We call such systems recursive. Secondly, to rewrite with S one has to be able to check if for a given u ∈ Γ∗ there is a rule → r ∈ S with = u, so we assume that the set L(S) of left-hand sides of the rules in S is a recursive subset of Γ∗ . Systems satisfying these two conditions are termed eﬀective rewriting systems. Clearly, every ﬁnite system is eﬀective. Notice also, that every recursive non-length-increasing system S (i.e., || ≥ |r| for every rule → r ∈ S) is eﬀective. Indeed, given u ∈ Γ∗ one can check if a rule u → v is in S or not for all words v with |v| ≤ |u|, thus eﬀectively verifying whether u ∈ L(S) or not. The argument above shows that for a recursive non-length-increasing system S one can eﬀectively enumerate all the rules in S in such a way 0 → r0 , 1 → r1 , . . . , i → ri , . . .

(1)

that if i < j then i j in the length-lexicographical ordering and also if i = j then ri rj . We call this enumeration of S standard. Proposition 2.10. Let S be an inﬁnite eﬀective convergent system. Then the word problem in the monoid MS deﬁned by S is decidable. Proof. Given a word u ∈ Γ∗ one can start the rewriting process by applying rules from S. Indeed, for a given factor w of u one can check if w ∈ L(S) or not, thus, enumerating all factors of u, one can either ﬁnd a factor w of u with w ∈ L(S) or prove that u is S-irreducible. If such w exists one can enumerate all pairs (w, v) with v ∈ Γ∗ and check one by one if (w, v) ∈ S or not. This procedure eventually terminates with a rule w → v ∈ S. Now one can apply this rule to u and rewrite u into u1 . Applying again this procedure to u1 one eventually arrives at a unique S-irreducible word u . To check if two words are equal in the monoid MS one can ﬁnd their S-irreducibles and check whether they are equal or not. There are various modiﬁcations of the algorithm described above that work for other types of, not necessarily convergent, inﬁnite systems. We consider some of these below.

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3. Length-reducing and Dehn systems 3.1. Finite length-reducing systems In this section we study a very particular type of rewriting system, called lengthreducing, where, for every rule → r one has || > |r|. The main interest in length-reducing systems comes from the fact that, contrary to the case of ﬁnite convergent systems, the algorithm for computing the reduced forms is fast. Lemma 3.1. [7] If S is a ﬁnite length-reducing string rewriting system then irreducible descendants of a given word can be computed in linear time (in the length of the word). This result is well known, we use it in many parts of the paper, and it can be seen easily as follows. Proof. First, we choose some ε > 0 such that (1 − ε)|| ≥ |r| for all rules (, r) ∈ S. Now, consider an input w ∈ Γ∗ of length n = |w|. For the moment let a ∗ conﬁguration be a pair (u, v) such that (i) w ⇐⇒ uv and (ii) u is irreducible. The S

goal is to transform the initial conﬁguration (1, w) in O(n) steps into some ﬁnal conﬁguration (w, ˆ 1). Say we are in the conﬁguration (u, v). The goal is achieved if v = 1. So assume that v = av where a is a letter. If ua is irreducible then we replace (u, av ) by (ua, v ), and (ua, v ) is the next conﬁguration. If however ua is reducible then we can write ua = u for some (, r) ∈ S; and u is irreducible. So, we replace (u, av ) by (u , rv ), and (u , rv ) is the next conﬁguration. The algorithm is obviously correct. Deﬁning the weight γ of conﬁgurations by γ(u, v) = (1 − ε)|u| + |v| we see that γ reduces from one conﬁguration to the next by at least ε. Hence we have termination in linear time. In fact, length reducing rewriting systems arise naturally in the class of small cancellation groups and more generally hyperbolic groups, which we might regard as a paradigm for groups with easily solvable word problem. To be precise: a group G is hyperbolic if and only if there is a ﬁnite generating set Γ for G and a ﬁnite length-reducing system S ⊆ Γ × Γ (so G = Γ/S) such that a word w represents the trivial element of G if and only if w can be S-reduced to the empty word, see [2]. In other words a group is hyberbolic if and only if there exists a ﬁnite length-reducing system which is conﬂuent on the empty word. Deﬁnition 3.2. A length-reducing string rewriting system which is conﬂuent on the empty word is called a Dehn system. If a group is deﬁned by a ﬁnite length-reducing Dehn rewriting system then the rewriting algorithm is known in group theory as the Dehn algorithm. More general deﬁnitions of Dehn algorithms, for rewriting systems over a larger alphabet than the generators of the group, have been studied by Goodman and Shapiro [25] and Kambites and Otto [32]. In particular in [25] it is shown that such generalised

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Dehn algorithms solve the word problem in ﬁnitely generated nilpotent groups and many relatively hyperbolic groups. It is known that, given a ﬁnite presentation of a hyperbolic group G, one can produce a ﬁnite Dehn presentation of G by adding, to a given presentation, all new relators of G up to some length (which depends on the hyperbolicity constant of G). However, this algorithm is very ineﬃcient and the following questions remain. Problem 3.3. Is there a Knuth-Bendix type completion process that, given a ﬁnite presentation of a hyperbolic group G, ﬁnds a ﬁnite Dehn presentation of G? Problem 3.4. Is there an algorithm that, given a ﬁnite presentation of a hyperbolic group, determines whether or not this presentation is Dehn? Notice that some partial answers to this question are known. Namely, in [3] Arzhantseva has shown that there is an algorithm that, given a ﬁnite presentation of a hyperbolic group and α ∈ [3/4, 1), detects whether or not this presentation is an α-Dehn presentation. Here a presentation X | R of a group G is called an α-Dehn presentation if any non-empty freely reduced word w ∈ (X ∪ X −1 )∗ representing the identity in G contains as a factor a word u which is also a factor of a cyclic shift of some r ∈ R±1 with |u| > α|r|. 3.2. Inﬁnite length-reducing systems Let us discuss some algorithmic aspects of rewriting with inﬁnite length-reducing systems. Proposition 3.5. Let S ⊆ Γ∗ × Γ∗ be an inﬁnite recursive string rewriting system. Then the following hold. 1) If S is length-reducing then an irreducible descendant of a given word can be computed. 2) If S is Dehn and MS is a group, then the word problem in MS is decidable. Proof. The system S is eﬀective since it is recursive and length-reducing (see remark before Proposition 2.10). Now the argument in the proof of Proposition 2.10 shows that for a given w one can eﬀectively ﬁnd an S-irreducible representative of w, so 1) and 2) follow. In the case of length-reducing systems one can try to estimate the time complexity of the algorithms involved. To this end we need the following deﬁnition. Let S be an eﬀective non-length increasing rewriting system and 0 → r0 , 1 → r1 , . . . , i → ri , . . . its standard enumeration (see Section 2.4). If there is an algorithm A and a polynomial p(n) such that for every n ∈ N the algorithm A writes out the initial part of the standard enumeration of S with |i | ≤ n in time p(n) then the system S is called enumerable in time p(n) or Ptime enumerable. In particular, we say that S is linear (quadratic) time enumerable if the polynomial p(n) is linear (quadratic).

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Proposition 3.6. Let S ⊆ Γ∗ ×Γ∗ be an inﬁnite non-length increasing string rewriting system, which is enumerable in time p(n). Then the following hold. 1) If S is length-reducing then an irreducible descendant of a given word w can be computed in polynomial time. 2) If S is Dehn and MS is a group then the word problem in MS is decidable in polynomial time. Proof. Given a word w one can list in time p(|w|) all the rules → r of the standard enumeration of S with || ≤ |w|. Now in time O(p(|w|)|w|2 ) one can check whether one of the listed rules can be applied to w or not. This proves 1) and 2). 3.3. Weight-reducing systems Many results above can be generalised to weight-reducing systems. A weight γ assigns to each generator an a positive integer γ(a) with the obvious extensions to words by γ(a1 · · · an ) = i=1 γ(ai ). A system is called weight-reducing if, for every rule → r, one has γ() > γ(r). The following statements in this paragraph are taken from [16]. It is decidable whether a ﬁnite system is weight-reducing by linear integer programming. The reason to consider weight-reducing systems is that there ∗ are monoids like { a, b, c } /ab = c2 having an obvious ﬁnite convergent weightreducing presentation, but where no ﬁnite convergent length-reducing presentation exists. For groups the situation is unclear. Actually, the following conjecture has been stated. Conjecture 3.7. Let G be a ﬁnitely generated group. Then the following assertions are equivalent. 1) G is a plain group, i.e., G is a free product of free and ﬁnite groups. 2) G has a ﬁnite convergent length-reducing presentation. 3) G has a ﬁnite convergent weight-reducing presentation. The implications 1) =⇒ 2) =⇒ 3) are trivial, and 2) =⇒ 1) is known as the Gilman conjecture and was stated ﬁrst in [23]. It is clear that the conjugacy problem can be decided in plain groups and this holds for groups G having a ﬁnite weight-reducing presentation too. In convergent fact, for s, t ∈ G the set Rs,t = g ∈ G gsg −1 = t is an eﬀectively computable rational subset of G.

4. Preperfect systems 4.1. General results In this section we discuss preperfect rewriting systems, which play an important part in solving their word problem and ﬁnding geodesics (shortest representatives in the equivalence classes) in groups.

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Deﬁnition 4.1. A Thue system is a rewriting system S ⊆ Γ∗ × Γ∗ such that the following conditions hold. i) If −→ r ∈ S then || ≥ |r|. ii) If −→ r ∈ S, with || = |r|, then r −→ ∈ S too. To every rewriting system is associated an equivalent Thue system. In order to specify a Thue system which is equivalent to a rewriting system S one can do the following: symmetrize S by adding all the rules r −→ whenever −→ r ∈ S, then throw out all the length increasing rules. The new system, denoted T (S), is called the Thue resolution of S. It follows that every monoid has a Thue presentation. Deﬁnition 4.2. A conﬂuent Thue system is called preperfect. The main interest in preperfect systems in algebra comes from the following known (and easy) complexity result: for which we require the following deﬁnition. Deﬁnition 4.3. A word w ∈ Γ∗ is termed S-geodesic, with respect to a string ∗ rewriting system S, if it has minimal length in its ⇐⇒-equivalence class (and S

simply geodesic where no ambiguity arises). Clearly, S-geodesic words are precisely the geodesic words in the monoid Γ/S relative to the generating set Γ, i.e., they have minimal length among all the words in Γ∗ that represent the same element in Γ/S. Sometimes we say that a ∗ word w ∈ Γ∗ is a geodesic of a word u ∈ Γ∗ if w is S-geodesic and ⇐⇒-equivalent S to u. Proposition 4.4. If a rewriting system S is ﬁnite and preperfect, then one can decide the word problem in the monoid deﬁned by S in polynomial space and hence in exponential time. Moreover, along the way one can ﬁnd an S-geodesic of a given word w as well as all S-geodesics of w. A locally conﬂuent (strictly) length-reducing system is convergent, hence, from the above, preperfect. However the Thue resolution of an arbitrary ﬁnite convergent rewriting system may fail to be terminating or conﬂuent as simple examples show. (Let Γ = {a, b, c, d, u, v} and S be the system with rules ab −→ u, bc −→ v, uc −→ d3 and av −→ d3 . Then T (S) is not conﬂuent. The system with one rule a −→ b has non-terminating Thue resolution.) It is also easy to see that T (S) may be preperfect when S is not conﬂuent. On the other hand, if a conﬂuent system S has no length-increasing rules then the Thue resolution can be constructed by symmetrizing S relative to all length preserving rules in S (by adding the rule r −→ for each length preserving rule −→ r ∈ S) and a straightforward argument shows that in this case T (S) is conﬂuent, so preperfect. Lemma 4.5. If S is a conﬂuent rewriting system with no length-increasing rules then the Thue resolution T (S) is preperfect.

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For a system S (for example a Thue system) where all rules → r ∈ S are either length-reducing || > |r|, or length-preserving || = |r|, it is convenient to split S into a length reducing part SR and a length preserving part SP , so S = SR ∪ SP . If S is a Thue system then all S-geodesic words that lie in the same equivalence class have the same length and any two of them are SP -equivalent (can be transformed one into other by a sequence of rules from SP ). Therefore, the word length in Γ∗ induces a well-deﬁned length on the factor-monoid M = Γ∗ /SP (application of relations from SP does not change the length). Hence, one can view SR as a length reducing rewriting system over the monoid M = Γ∗ /SP , in which case we assume that SR ⊆ M × M . Note that if SP is ﬁnite and if there is an eﬀective way to perform reduction steps with SR then the word problem in M is decidable. Decidability of the word problem in M = Γ∗ /SP allows one to test whether a given rule from SR is applicable to an element of M . Since SR ⊆ M × M is terminating it suﬃces to show local conﬂuence to ensure convergence. This may tempt one to introduce an analogue of the Knuth-Bendix completion. However, in general an inﬁnite number of critical pairs may appear in the Knuth-Bendix process, and one needs to be able to recognize when the current system becomes preperfect. Unfortunately, this is algorithmically undecidable. More precisely, the following result holds. Theorem 4.6 ([38]). The problem of verifying whether a ﬁnite Thue system is preperfect or not is undecidable. In fact in [39] this problem is shown to be undecidable even in the case of a Thue system whose length-preserving part SP consists only of a single rewriting rule of the form ab ←→ ba. On the other hand, under some additional assumptions such a procedure can yield useful results ([17, 18]) – good examples in our context are graph groups, c.f. Section 7.1. In the ﬁnal part of this section we discuss some complexity issues in computing with preperfect systems. By Proposition 4.4 ﬁnite preperfect systems allow one to solve the word problem and ﬁnd geodesics in at most exponential time. Proposition 4.7. Let S be an inﬁnite preperfect rewriting system. Then: 1) if S is recursive then the word problem in the monoid MS deﬁned by S is decidable; 2) if S is Ptime enumerable then one can solve the word problem in MS and ﬁnd a geodesic of a given word in exponential time. Proof. Since preperfect rewriting systems are non-length-increasing it follows that recursive preperfect systems are eﬀective (see the remark before Proposition 2.10). Therefore, given a word w one can eﬀectively list all the rules in S with the lefthand sides of length at most |w|. Denote this subsystem of S by Sw . Now rewriting w using S is exactly the same as using Sw , so 1) and 2) follow from the argument in the proof of Proposition 4.4 for ﬁnite preperfect systems.

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5. Geodesically perfect rewriting systems In this section we consider a subclass of Thue systems which are designed to deal with geodesics in groups or monoids. In particular, we study conﬂuent geodesic systems, which form a subclass of preperfect string rewriting systems, and which behave better in many ways than general preperfect systems. We call these systems geodesically perfect, as this indicates their essential properties and ﬁts with the terminology of preperfect systems. However as discussed in Section 1 they are also known in the literature as almost conﬂuent or quasi-perfect. The motivation for the study of geodesically perfect systems in group theory comes mainly from attempts to solve the, algorithmically diﬃcult, geodesics problem: that is, given a ﬁnite presentation of a group G and a word w in the generators, ﬁnd a word of minimal length representing w as an element of G. 5.1. Geodesic systems We consider ﬁrst a somewhat larger, less well-behaved, class of rewriting systems. Deﬁnition 5.1. A string rewriting system S ⊆ Γ∗ × Γ∗ is called geodesic if Sgeodesic words are exactly those words to which no length reducing rule from S can be applied. Note that if S is a geodesic rewriting system then its Thue resolution is also geodesic. This allows us to assume, without loss of generality, that geodesic systems are Thue systems. Remark 5.2. Dehn rewriting systems are not in general geodesic: they need only rewrite words that represent the identity to (empty) geodesics in Γ∗ /S. A ﬁnite geodesic system gives a linear time algorithm to ﬁnd a geodesic of a given word u ∈ Γ∗ . The following algebraic characterisation of ﬁnite geodesic systems in groups is given in [24]. (The deﬁnition of geodesic in [24] is slightly more restrictive than ours, however this makes no diﬀerence to the result.) Theorem 5.3 ([24]). A group G is deﬁned by a ﬁnite geodesic system S if and only if G is a ﬁnitely generated virtually free group. From the result of Rimlinger quoted above ﬁnitely generated virtually free groups are precisely the universal groups of ﬁnite pregroups. It follows that every ﬁnite length reducing geodesic system can be transformed to the length reducing part of the rewriting system (see Section 8.1) associated with a ﬁnite pregroup. The following result follows from Proposition 3.6. Proposition 5.4. Let S be a geodesic Ptime enumerable string rewriting system such that the monoid MS is a group. Then the word problem in the group MS is decidable in polynomial time. Very little is known about geodesic systems which do not present groups. In particular, it is not clear whether the word problem remains decidable: that is, ∗ given u, v ∈ Γ∗ decide whether or not u ⇐⇒ v. S

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Problem 5.5. Does there exist a ﬁnite geodesic system S for which the word problem is undecidable? The following result demonstrates one of the principal diﬃculties of working with geodesic systems. Theorem 5.6. It is undecidable whether a ﬁnite rewriting system is geodesic. Proof. The proof is a modiﬁcation of the proof by Narendran and Otto [39] which showed undecidability of preperfectness in presence of a single commutation rule. We need some notation and we adhere as far as possible to that of [39]. We shall deﬁne the computation of a Turing machine by a set of rewriting rules. A conﬁguration of the machine is then a particular form of word over the tape alphabet, the states and the end markers. In detail let Σ be a ﬁnite set, the tape alphabet, let Σ be a disjoint copy of Σ, let Q be a ﬁnite set of states, and α and β be special symbols representing end markers. There are two marked states q0 and qf , the initial and ﬁnal states. The computation of the machine can be described by a ﬁnite set of rules which fall into the following categories, where we use the notation p, q ∈ Q, p = qf , a, a , b ∈ Σ: 1) pa −→ a q. (Read a in state p, write a , move one step to the right, switch to state q.) 2) bpa −→ qba . (As above, but move one step to the left.) 3) pβ −→ paβ. (Create new space before the right end marker.) These rewriting rules constitute the rewriting system associated to M . We assume that the machine is deterministic, so there are no overlapping rules. A conﬁguration ∗ of a (deterministic) Turing machine is then a word αuqvβ with u ∈ Σ , v ∈ Σ+ , and q ∈ Q. The initial conﬁguration on input x ∈ Σ∗ is the word αq0 xβ. We assume that the machine stops if and only if it reaches the state qf . Now let M be a Turing machine for which it is undecidable whether or not computation halts on input x ∈ Σ∗ . Using this machine we are going to construct, for each x ∈ Σ∗ , a new length reducing rewriting system Sx , which is geodesic if and only if the machine M does not stop on input x. The alphabet Γ of each such system is to consist of the symbols of Σ ∪ Σ ∪ Q ∪ {α, β} and new additional symbols d, e, γ, δ, I, C. The system Sx will consist of rules which simulate the computation of M on input x, with some additional control on the number of steps of the computation carried out. Let x ∈ Σ∗ . To begin with, we introduce rules leading to two diﬀerent initial conﬁgurations. Let m = |x| + 5. We deﬁne the two rules αq0 xβγ ←− IC m −→ αq0 xβδ. (1)

(2)

Next we introduce rules, involving d, e, γ and δ, to control the number of steps of the simulation. Symbols γ and δ convert d’s to e’s. The latter act as tokens to control the number of steps performed by the simulation of M . Both γ and δ move right consuming three d’s and producing two e’s, the diﬀerence being that γ may

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move to the right arbitrarily far from β whereas δ is forced to remain very close to β. The eﬀect of each rule on the length of a word is the same. Explicitly, we add new rules of the form: γddd −→ eeγ,

βδddd −→ βeeδ.

Note that, using rule (1), all words in IC m (ddd)∗ now reduce as follows ∗

IC m d3n =⇒ αq0 xβγd3n =⇒ αq0 xβe2n γ. (1)

However, using the rule IC m −→ αq0 xβδ in the ﬁrst step we can only do (2)

m 3n

IC d

∗

=⇒ αq0 xβδd3n =⇒ αq0 xβeeδd3n−3 (2)

and then, for n ≥ 2, we are stuck. Now we bring e into the game. The letter e is used to enable a computation step of M . It can move to the left until it is at distance one to the right of a state symbol. The generic rules for e allow e to move left and are as follows: abee −→ aeb,

aebee −→ aeeb for a ∈ Σ, b ∈ Σ ∪ {β}.

Let us describe the eﬀect of these rules on words of the form αupa1 · · · ak βδd3n ∗

where n is huge (and k is viewed as constant k ≥ 0), ai ∈ Σ, u ∈ Σ . The maximal possible reduction leads to a word of the form

αupa1 ee · · · ak eeβeeδd3n . In this case, if n is large enough, then n > 0, no further reduction is possible and actually n − n ∈ O(1). At this point we introduce rules to simulate the computation of the machine M . There is one simulation rule corresponding to each rule in the rewriting system associated to M . More precisely we introduce a rule uee −→ v for each rewriting rule u −→ v of M : so we have simulation rules of three types (where again we use the notation p, q ∈ Q, a, a , b ∈ Σ) paee −→ a q,

bpaee −→ qba ,

pβee −→ paβ.

The system Sx consists of the rules deﬁned so far, which we list in Figure 1, so is length reducing. Now assume that the machine M halts on input x. This implies that only ﬁnitely many computation steps t can be performed. Again choose n huge and view t and |x| as constants. Consider a word of the form IC m d3n . Starting a reduction with the second rule we get stuck at an irreducible word when the simulation reaches state qf : ∗

IC m d3n −→ αq0 xβδd3n =⇒ αuqf a1 ee · · · ak eeβeeδd3n (2)

72

V. Diekert, A.J. Duncan and A.G. Myasnikov I) Initial rules: αq0 xβγ ←− IC m −→ αq0 xβδ. (1)

(2)

II) Step control rules, for a ∈ Σ, b ∈ Σ ∪ {β}: γddd −→ eeγ, βδddd −→ βeeδ, abee −→ aeb, aebee −→ aeeb. III) Simulation rules, for a, b ∈ Σ, p ∈ Q\{qf }: paee −→ a q, bpaee −→ qba , pβee −→ paβ. Figure 1. The system Sx . at which point n − n ∈ O(1). The system Sx cannot be geodesic because with the other initial rule we can ﬁrst move Γ to the right of all the d’s thereby losing n letters immediately: ∗

IC m d3n −→ αq0 xβγd3n =⇒ αq0 xβd2n γ (1)

and then when the simulation reaches the state qf the resulting irreducible word will end e2n γ instead of δd3n (and otherwise will be the same). It remains to cover the case when the machine does not halt on input x. We shall show that in this case the system Sx is geodesic. Note that, as M never reaches state qf , for all n > 0 ∗

αq0 xβe2n =⇒ αupyβ, Sx

∗

where u ∈ Σ , p ∈ Q and y ∈ (Σ∪{e})∗ . For technical reasons we deﬁne a sequence of words wi for i ≥ 0 as follows. We let w0 = q0 x and let αwi+1 β be deﬁned to be the irreducible descendant of αwi βee. The sequence of words wi is inﬁnite because the machine does not stop on input x. Thus ∗

∀i ≥ 0 : αwi βee =⇒ αwi+1 β ∈ IRR(Sx ). Sx

Now we add inﬁnitely many rules to Sx to form a new system Tx as follows: ∀i ≥ 0 : αwi βδ −→ αwi βγ. As the rules of Tx are generated by steps of the Knuth-Bendix completion procedure applied to Sx the congruences generated by Sx and Tx are the same.

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IV.) Completion rules: ∀i ≥ 0 : αwi βδ −→ αwi βγ ∗

where αwi βee =⇒ αwi+1 β, ∀i ≥ 0, and w0 = q0 x. Sx

Figure 2. The additional rules of system Tx . To summarise, the system Tx consists of the rules of Figure 1 and those listed in Figure 2. Thus Tx is terminating and local conﬂuence can be checked directly. Each word w ∈ Γ∗ has a unique factorisation where we choose k and all nj to be maximal: u0 (IC m dn1 )u1 · · · (IC m dnk )uk . The beneﬁt of the system Tx is that it provides us with canonical geodesics. A geodesic of w is given by: u 0 (αwi1 βγdm1 ) u1 · · · (αwik βγdmk ) uk , where mj = nj mod 3. The crucial observation is that allowing only rules from Sx we achieve exactly the same form with the exception that some γ’s are still δ’s. Thus, the system is geodesic. 5.2. Geodesically perfect systems Deﬁnition 5.7. A string rewriting system S ⊆ Γ∗ ×Γ∗ is called geodesically perfect if i) S is geodesic and ∗ ∗ ii) if u, v ∈ Γ∗ are S-geodesics then u ⇐⇒ v if and only if u ⇐⇒ v, where SP is S

the length-preserving part of S.

SP

Again, it follows directly that if S is a geodesically perfect system then so is its Thue resolution, so we can assume that geodesically perfect systems are Thue. If S is a geodesically perfect Thue system then we write it as S = SR ∪ SP where SR is its length reducing part and SP its length preserving part. It also follows from the deﬁnition that a geodesically perfect system is conﬂuent. There is a simple procedure to describe geodesics of elements in the monoid Γ∗ /S deﬁned by a geodesically perfect Thue system S. Namely, the geodesics of a given word w ∈ Γ∗ are the SR -reduced forms of w and any two such geodesics can be obtained from one another by applying ﬁnitely many rules from SP . Moreover it is shown in [6] that the word problem for monoids deﬁned by ﬁnite geodesically perfect rewriting systems is PSPACE complete. The following result relates geodesically perfect to preperfect Thue systems. Proposition 5.8. Let S ⊆ Γ∗ × Γ∗ be a Thue system. Then 1) if S is geodesically perfect then it is preperfect and 2) if S is preperfect and geodesic then it is geodesically perfect.

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Proof. 1) follows from the observation that geodesically perfect implies conﬂuent. To see 2) observe that S is conﬂuent, hence Church-Rosser. Therefore, if u, v are ∗ ∗ ∗ two geodesics with u ⇐⇒ v then u =⇒ w and w ⇐= v for some w ∈ Γ∗ . Since u, v S

S

S

∗

∗

S

S

are S-geodesics the only rules that could be applied in u =⇒ w and w ⇐= v are ∗

length preserving, hence u ⇐⇒ v, as required.

SP

In Section 8.1 we will describe a general tool to construct geodesically perfect systems deﬁning groups: based on the fact that rewriting systems associated with pregroups are always geodesically perfect. In Corollary 8.7 we prove that groups deﬁned by ﬁnite geodesic systems are exactly the groups deﬁned by ﬁnite geodesically perfect systems. Obviously, every geodesic rewriting system S contains the length-reducing part TR of some (inﬁnite) geodesically perfect Thue system T deﬁning the same monoid. Indeed, one can obtain T by ﬁrst constructing the Thue resolution T of S and then adding length-preserving rules to T to make it conﬂuent. But it is not true that every ﬁnite geodesic rewriting system S is the length-reducing part of a ﬁnite geodesically perfect system deﬁning the same monoid. To see this consider the following example. Example 5.9. The following system is geodesic, and it is not the length-reducing part of any ﬁnite geodesically perfect system deﬁning the same quotient monoid. add −→ ab,

add −→ ac,

bdd −→ eb,

cdd −→ ec.

Indeed, let S be the system above, and let T = S ∪ { b ←→ c }. The new system T is geodesically perfect by Proposition 6.1. But T -geodesics are computed by using ∗ ∗ rules from S. As ⇐⇒ ⊆ ⇐⇒ we see that S is a geodesic system. S

T

Let us show that S is not the length-reducing part of any equivalent, ﬁnite, geodesically perfect system. For a contradiction, assume that a ﬁnite set T of nontrivial symmetric rules can be added to S such that S ∪ T becomes geodesically perfect and is equivalent to S. Assume T involves a new letter, say f . Then f is equal to some word uf over {a, b, c, d, e} which is irreducible with respect to S. If uf ∗ is empty then we do not need f , hence uf is nonempty and we have uf =⇒ f . The T

rules of T are symmetric (hence length preserving), so f is accompanied by a rule, say f ←→ a, and f is redundant. So, actually we may assume T ⊆ {a, b, c, d, e}∗ × ∗ ∗ ∗ {a, b, c, d, e}∗. Clearly, aen b ⇐= ad2n+2 =⇒ aen c, hence aen b ⇐⇒ aen c and so aen b S

S

S

and aen c are in the same class and are S-reduced. Because T is ﬁnite, some lefthand side of T must contain a word u ∈ ae∗ ∪ e∗ ∪ e∗ b ∪ e∗ c. But all these words u are S-reduced, hence geodesic. Moreover, for any such u there is no other word v in the same class as u and of the same length. So, for large enough n the rules of T cannot be applied to either aen b or aen c. As T is the length preserving part of the supposedly geodesically perfect system S ∪ T , this is the required contradiction.

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Remark 5.10. Let M = {a, b, c, d, e}∗/S be the quotient monoid as in Example 5.9. The proof above can be modiﬁed in order to show that actually there is no ﬁnite system T ⊆ {a, b, c, d, e}∗ × {a, b, c, d, e}∗ which is geodesically perfect and which deﬁnes M . However, if we use an additional letter f then the following system deﬁnes M , too. dd −→ f,

af ←→ ab,

af ←→ ac,

bf ←→ eb,

cf ←→ ec.

The system is geodesically perfect, by Proposition 6.1 again. We note that Example 5.9 illustrates a general fact: namely that if S is a rewriting system and there exists a set T of symmetric rules such that S ∪ T is geodesically perfect (but not necessarily equivalent to S) then S itself is geodesic. Since geodesic systems are undecidable whereas geodesically perfect systems are decidable this could prove to be a useful test for a geodesic system.

6. Knuth-Bendix completion for geodesically perfect systems A classical result of Nivat and Benois (stated in Proposition 6.1) shows that it is decidable whether a ﬁnite Thue system is geodesically perfect. In order to explain the criterion we need the notion of critical pair. All rewriting systems S in this subsection are viewed as Thue systems and split into a length reducing part SR and a length preserving part of symmetric rules SP . By deﬁnition, a critical pair is a pair (x, y) arising from the situation x ⇐= z =⇒ y ( 1 ,r1 )

( 2 ,r2 )

subject to the following conditions. 1. (1 , r1 ) ∈ SR is length reducing but (2 , r2 ) ∈ S can be any rule. 2. z = i ui = uj j , with |ui | < |j | and i, j ∈ {1, 2} such that i = j, implies ui = uj = 1. Proposition 6.1 ([41]). A ﬁnite Thue system S is geodesically perfect if and only if, for all critical pairs (x, y), there are words x and y such that with length reducing reductions we have: ∗ ∗ x ⇐= x, y =⇒ y , SR

SR

and with length preserving reductions we have: ∗

x ⇐⇒ y . SP

Proof. The proof is not very diﬃcult and can be found, for example, in the book [6, Thm. 3.6.4]. Remark 6.2. Note that the words x and y in Proposition 6.1 need not be irreducible w.r.t. the length reducing subsystem SR . This fact is actually used in the proof of Proposition 6.3.

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This criterion leads to the following version of the Knuth-Bendix procedure. Consider a ﬁnite Thue system S0 . We shall construct a series of Thue systems S0 ⊆ S1 ⊆ S2 ⊆ · · · such that the union over all Si is geodesically perfect and we have Si = Si+1 (i.e., the completion procedure stops) if and only if there exists a ﬁnite Thue system T which is geodesically perfect and equivalent to S0 , that is ∗ ∗ ⇐⇒ = ⇐⇒ . We divide the procedure into phases. We assume that in phase i S0

T

a Thue system Si = SR ∪ SP has been deﬁned such that SR contains the length ∗ ∗ reducing rules, SP contains the length preserving rules, and ⇐⇒ = ⇐⇒ . S0

Si

We begin phase i + 1 by computing a list of all critical pairs of the system Si (which were not already considered in phases 1 to i). For each such pair (x, y) choose words x , y, irreducible with respect to the subsystem SR , such that ∗

∗

x ⇐= x,

y =⇒ y.

SR

SR

Deﬁne new rules as follows. • If | x| > | y | then add the rule x −→ y to SR . • If | y | > | x| then add the rule y −→ x to SR . • If | y | = | x| then test whether or not ∗

x ⇐⇒ y. SP

If the answer is negative then add the symmetric rule x ←→ y to SP . The system Si+1 is deﬁned to be Si together with all new rules which have been added to resolve all critical pairs of Si . On a formal level we deﬁne Si for all i ≥ 0 but, of course, the procedure stops as soon as Si = Si+1 , i.e., no new rules are needed to resolve critical pairs of Si . Thus, if it stops with Si = Si+1 then Si is a ﬁnite geodesically perfect Thue system, which is equivalent to S0 (and we have Si = Sj for all i ≤ j). However, what we really wish is stated in the following proposition. Proposition 6.3. Let S0 = SR ∪ SP be a ﬁnite Thue system with length reducing rules SR and length preserving rules SP . Let S 0 ⊆ S 1 ⊆ · · · Si ⊆ · · · be the sequence of Thue systems which are computed by the Knuth-Bendix comple tion as described above. Let S = i≥0 Si . Then the system S is geodesically perfect ∗

∗

∗

S0

Si

s

and we have ⇐⇒ = ⇐⇒ = ⇐⇒ for all i ≥ 0. Moreover the following statements are equivalent. 1) We have Si = Si+1 for some i ≥ 0. 2) The Thue system S is ﬁnite and geodesically perfect. 3) There exists some ﬁnite geodesically perfect Thue system T such that ∗

∗

S0

T

⇐⇒ = ⇐⇒ .

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Proof. If Si = Si+1 , for some i ≥ 0, then clearly S is ﬁnite. It is geodesically perfect by the criterion of Nivat and Benois, cf. Proposition 6.1. So, assume there ∗ ∗ exists some ﬁnite geodesically perfect Thue system T with ⇐⇒ = ⇐⇒ . We S0

T

have to show that the procedure stops. We let m be large enough that m ≥ max { || | (, r) ∈ T }. Next we consider i large enough that Si contains all rules from S where the left-hand side has length of at most m. Clearly, an index i ∈ N with this property exists. We will show that Si is geodesically perfect, and Proposition 6.1 immediately implies Si = Si+1 . For technical reasons, in a ﬁrst step we remove from T all length preserving rules (, r) ∈ T where we can apply to a length reducing rule of T . It is clear that the new and smaller system T ∗ ∗ is still geodesically perfect and ⇐⇒ = ⇐⇒ ; so we replace T with T . Since S0

T

now (, r) ∈ T with || = |r| implies that and r are geodesics and since S is ∗ ∗ geodesically perfect and m is large enough, we see that ⇐⇒ ⊆ ⇐⇒ . TP

(Si )P

Next consider some word x which is irreducible with respect to the length is a geodesic. Indeed assume the contrary. reducing rules in Si . The claim is that x Then a length reducing rule (, r) ∈ T can be applied to x . Since is not geodesic, there is a length reducing rule in S which can be applied to but due to the deﬁnition of m this rule is in Si , too. Thus, we have a contradiction and so Si is ∗ ∗ geodesic. Now suppose that x and y are geodesic and that x ⇐⇒ y. Then x ⇐⇒ y ∗

∗

∗

∗

TP

TP

(Si )P

(Si )P

Si

⇐⇒ y so Si is geodesically perfect. so x ⇐⇒ y. As ⇐⇒ ⊆ ⇐⇒ this implies x

T

A ﬁnite geodesic (or geodesically perfect) rewriting system S ⊆ Γ∗ ×Γ∗ allows one to ﬁnd S-geodesics in linear time. In particular, if the monoid M = Γ∗ /S deﬁned by S is a group one can solve the word problem in M in linear time. However, in general there seems to be no linear time reduction from the word problem in a monoid M to the geodesic problem.

7. Examples of preperfect systems in groups 7.1. Graph groups Let Δ = (Σ, E) be an undirected graph. The graph group (or right angled Artin group, or partially commutative group) deﬁned by Δ is the group G(Δ) given by the presentation G(Δ) = F (Σ)/ { ab = ba | (a, b) ∈ E } , where F (Σ) is the free group with basis Σ. The group G(Δ) has a monoid presentation given by a preperfect rewriting system SΔ . Indeed, let Γ = Σ ∪ Σ where

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Σ is a disjoint copy of Σ. The rules of SΔ are: aa ab

−→ 1 ←→ ba

if

(a, b), (a, b), (a, b), (a, b) ∩ E = ∅

where a, b ∈ Γ and a = a for all a ∈ Γ. If the graph Δ is ﬁnite the system SΔ provides us with a decision algorithm for solving the word problem in G(Δ), though not the fastest one (WP in graph groups can be solved in linear time, see [53, 18]). However, the system SΔ is very intuitive and simple and it gives the geodesics in G(Δ), which are precisely the words whose length cannot be reduced by SΔ . Although it is preperfect the system SΔ is not geodesically perfect. However every graph group may be constructed by a sequence of HNN-extensions and free products with amalgamation, starting with inﬁnite cyclic groups, and so, from the results of Section 8 below, it follows that these groups may be deﬁned by (inﬁnite) geodesically perfect systems. Moreover ﬁnite convergent rewriting systems for these groups have been found by Hermiller and Meier [27] (see also [5, 22, 52]). 7.2. Coxeter groups Let D3 = {a, b}∗ /{a2 = 1, b2 = 1, (ab)3 = 1} be a dihedral group. Deﬁne a preperfect system S by the following rules aa −→ 1, bb −→ 1, aba ←→ bab. More generally, a Coxeter group on n generators a1 , . . . , an is given by a symmetric n × n matrix (mij ) with entries in N and 1’s on the diagonal. The deﬁning relations are given by: (ai aj )mij = 1

for all 1 ≤ i, j ≤ n.

a2i

= 1 since mii = 1; and if mij = 0 then the equation Note that this implies (ai aj )0 = 1 is trivial. (Therefore it is also common to write (ai aj )∞ = 1, because ai aj turns out to be an element of inﬁnite order in this case.) The word problem of Coxeter groups can be solved by the preperfect Tits system [51] (see also ([9, 1, 12]) of rewriting rules: a2i −→ 1, (ai aj ai aj · · · ) ←→ (aj ai aj ai · · · )

for 1 ≤ i ≤ n, for 1 ≤ i, j ≤ n and |(ai aj ai aj · · · )| = |(aj ai aj ai · · · )| = mij .

The classical proof that this system is preperfect relies on the fact that Coxeter groups are linear [4]. Of course this system is not geodesically perfect. For virtually free Coxeter groups Corollary 8.7 guarantees the existence of a ﬁnite geodesically perfect rewriting system. It is shown in [26] that every Coxeter group is either virtually free or contains a surface group; but the question of whether the

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latter can be deﬁned by a geodesically perfect system (necessarily inﬁnite) remains open. Convergent rewriting systems for Coxeter groups have been constructed, using the Knuth-Bendix procedure, by le Chenadec [34], but in general these are not ﬁnite. Finite convergent rewriting systems for certain classes of Coxeter groups have been found by Hermiller [28] (see also [19, 8]). 7.3. HNN-extensions Let G be any group with isomorphic subgroups A and B. Let Φ : A → B an isomorphism and let t be a fresh letter. By G, t we mean the free product of G with the free group F (t) over t. The HNN-extension of G by (A, B, Φ) is the quotient group HNN(G; A, B, Φ) = G, t / t−1 at = Φ(a) a ∈ A . There is a normal form theorem for elements in HNN(G; A, B, Φ), which implies that G embeds into HNN(G; A, B, Φ) and shows under which restrictions decidability of the word problem for G transfers to HNN-extensions. Usually the normal form theorem is shown by appeal to a combination of arguments of Higman, Neumann and Neumann and Britton, see [35, Chapter IV, Theorem 2.1]. Another option is to deﬁne a convergent string rewriting system. To see this, let Γ = t, t−1 ∪ G \ {1} and view Γ as a possibly inﬁnite alphabet. We identify 1 ∈ G with the empty word 1 ∈ Γ∗ . We choose transversals for cosets of A and B. This means we choose X, Y ⊆ G such that there are unique decompositions G = AX = BY. We may assume that 1 ∈ X ∩ Y . The system S ⊆ Γ∗ × Γ∗ is now deﬁned by the following rules with the convention that [gh] denotes gh ∈ G (as a single letter or the empty word). t−1 t tg t−1 g

−→ 1; tt−1 −→ 1; gh −→ [gh], for all g, h ∈ G; −→ aty, if a ∈ A, a = 1, y ∈ Y, Φ(a)y = g in G; −→ bt−1 x, if b ∈ B, b = 1, x ∈ X, Φ−1 (b)x = g in G.

Proposition 7.1. The system S above is convergent and deﬁnes the HNN-extension of G by (A, B, Φ). Every irreducible normal form admits a unique decomposition as g = g0 tε1 g1 · · · tεn gn with n minimal such that n ≥ 0, g0 ∈ G \ {1}, and either εi = −1 with gi ∈ X or εi = 1 with gi ∈ Y , for all 1 ≤ i ≤ n. Proof. Obviously, Γ∗ /S deﬁnes the HNN-extension of G by by (A, B, Φ). Although the system has length-increasing rules it is not too diﬃcult to prove termination. Local conﬂuence is straightforward, so S is indeed convergent. Since all elements of G are irreducible we see that G embeds into the HNN-extension. Moreover, it is also clear that we obtain the normal form as stated in the proposition.

80

V. Diekert, A.J. Duncan and A.G. Myasnikov This convergent system also leads to the following well-known classical fact.

Corollary 7.2. Assume that have the following properties: H is ﬁnitely generated and has a decidable word problem, membership problems for A and B are solvable, and the isomorphism Φ : A → B is eﬀectively calculable. Then the HNN-extension of G by by (A, B, Φ) has a decidable word problem. Proof. We may represent all group elements in H by length-lexicographic ﬁrst elements (i.e., choose among all geodesics the lexicographical ﬁrst one). The transversal X (resp. Y ) may be chosen to consist of the length-lexicographic ﬁrst element of each coset Ag (resp. Bg), where g runs over G. Given g we can compute the representative of Ag in X (resp. Bg in Y ), because membership is decidable for A and B. Now, given b ∈ B, the ability to compute Φ allows us to ﬁnd a ∈ A with Φ(a) = b. Thus, all steps in computing normal forms are eﬀective. It should be clear however that the purpose of the system S above is not to decide the word problem eﬀectively; but rather to facilitate straightforward proofs of other results, such as Britton’s lemma. Consider the following system B of Britton reduction rules. t−1 t −1

t

at −→

−1

tbt

−→ −→

1;

tt−1 −→ 1;

Φ(a)

if a ∈ A;

Φ

−1

gh −→ f, if gh = f in G;

(b) if b ∈ B. ∗

The system B is length reducing, but not conﬂuent. However, =⇒ ⊆ =⇒, B

H

hence we can think of B as as subsystem of H. Britton’s lemma says that B is conﬂuent on all words which represent 1 in the HNN-extension. Here is a proof using our system S. Consider any Britton reduced word g. It has the form g = g0 tε1 g1 · · · tεn gn . Applying rules from H does not destroy the property of being Britton reduced and neither t nor t−1 can vanish. Thus, if g reduces to the empty word using H, then g is already the empty word. Observe that B is not a geodesic system, because atΦ(a)−1 is Britton reduced, but atΦ(a)−1 = t. In Example 8.2 below we construct a geodesically perfect rewriting system for an HNN-extension. 7.4. Free products with amalgamation There is a natural convergent (resp. geodesically perfect) rewriting system which deﬁnes amalgamated products. Let A and B be groups intersecting in a common subgroup H. This time we choose transversals for cosets of H in A and in B; that is X ⊆ A and Y ⊆ B with 1 ∈ X ∩ Y such that there are unique decompositions A = HX and B = HY . We let Γ = (A ∪ B) \ {1} and we identify 1 with the empty word in Γ∗ . We use the convention of writing [ab] for the product ab whenever it is deﬁned. This means [ab] is viewed as a letter in Γ or [ab] = 1 and it is deﬁned if either a, b ∈ A or a, b ∈ B.

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The system S ⊆ Γ2 × ({1} ∪ Γ ∪ Γ2 ) is now deﬁned by the following rules: ab −→ [ab] ab −→ [ah]y ba −→ [bh]x

if [ab] is deﬁned, if 1 = a ∈ A, h ∈ H, b = y ∈ Y, and b = [hy], if 1 = b ∈ B, h ∈ H, a = x ∈ X, and a = [hx].

The system deﬁnes the amalgamated product G = A ∗H B. It is terminating by a length lexicographical ordering. Local conﬂuence follows by a direct inspection, whence convergence. Again we obtain the normal form theorem (cf. [36, Corollary 4.4.1]): every element g of G has a unique decomposition as g = [hg0 ]g1 · · · gn , where h ∈ H, gi is a non-trivial element of X ∪ Y and gi and gi+1 do not lie in the same factor. However, in practice we may not wish to compute transversals explicitly. So let us apply only length reducing rules ab −→ [ab] only until we end up with a word g = g0 · · · gn , to which no length reducing rule may be applied. Since we cannot apply length reducing rules to g we obtain that ∀0 ≤ i < n : gi ∈ A ⇐⇒ gi+1 ∈ B \ H ∧ gi ∈ B ⇐⇒ gi+1 ∈ A \ H. Further applications of the rules of S preserve this property. Thus, S is geodesically perfect, even if we use the length preserving rules only in the direction indicated above. Moreover, if we cannot apply length reducing rules to g = g0 · · · gn then we have g = 1 if and only if both n = 0 and g0 = 1.

8. Stallings’ pregroups and their universal groups We now turn to the notion of pregroups in the sense of Stallings, [49], [50]. A pregroup P is a set P with a distinguished element ε, equipped with a partial multiplication m : D → P , (a, b) → ab, where D ⊆ P × P , and an involution (or inversion) i : P → P , a → a−1 , satisfying the following axioms for all a, b, c, d ∈ P . (By “ab is deﬁned” we mean to say that (a, b) ∈ D and m(a, b) = ab.) (P1) aε and εa are deﬁned and aε = εa = a; (P2) a−1 a and aa−1 are deﬁned and a−1 a = aa−1 = ε; (P3) if ab is deﬁned then so is b−1 a−1 and (ab)−1 = b−1 a−1 ; (P4) if ab and bc are deﬁned then (ab)c is deﬁned if and only if a(bc) is deﬁned, in which case (ab)c = a(bc); (P5) if ab, bc, and cd are all deﬁned then either abc or bcd is deﬁned. It is shown in [29] that (P3) follows from (P1), (P2), and (P4), hence can be omitted. The universal group U (P ) of the pregroup P can be deﬁned as the quotient monoid U (P ) = Γ∗ / { ab = c | m(a, b) = c } ,

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where Γ = P \ {ε} and ε ∈ P is identiﬁed again with the empty word 1 ∈ Γ∗ . The elements of U (P ) may therefore represented by ﬁnite sequences (a1 , . . . , an ) of elements from Γ such that ai ai+1 is not deﬁned in P for 1 ≤ i < n: such sequences are called P -reduced sequences or reduced sequences. Since every element in U (P ) has an inverse, it is clear that U (P ) forms a group. If Σ is any set then the disjoint union P = {ε} ∪ Σ ∪ Σ, where Σ is a copy of Σ, yields a pregroup with involution given by ε = ε, a = a, for all a ∈ Σ, such that pp = ε, for all p ∈ P . In this case the universal group U (P ) is nothing but the free group F (Σ). The universal property of U (P ) holds trivially, namely the canonical morphism of pregroups P → U (P ) deﬁnes the left-adjoint functor to the forgetful functor from groups to pregroups. Stallings [49] showed that composition of the inclusion map P → P ∗ with the standard quotient map P ∗ → U (P ) is injective, where P ∗ is the free monoid on P . The ﬁrst step of his proof establishes reduced forms of elements of U (P ), up to an equivalence relation ≈ which, for completeness, we describe here. Deﬁne ﬁrst a binary relation ∼ on the set of ﬁnite sequences of elements of P by (a1 , . . . , ai , ai+1 , . . . , an ) ∼ (a1 , . . . , ai c, c−1 ai+1 , . . . , an ), provided (ai , c), (c−1 , ai+1 ) ∈ D. Then Stallings’ equivalence relation ≈ is the transitive closure of ∼. Guiding examples are again amalgamated products and HNN-extensions. Example 8.1. As in Section 7.4, let A and B be groups intersecting in a common subgroup H. Consider the subset P = A ∪ B ⊆ G = A ∗H B. Deﬁne a partial multiplication p · q in the obvious way; that is p · q is deﬁned if and only if either p, q are both in A or p, q both in B. Then P is a pregroup where D = A×A∪B ×B. We obtain the following geodesically perfect rewriting system (where the length is computed w.r.t. P , thus elements of P are viewed as letters). 1 −→ ε p · q −→ r if (p, q) ∈ D, pq = r ∈ G a · b ←→ ah · h−1 b if a ∈ A \ H, b ∈ B \ H, h ∈ H. Example 8.2. Let H be the HNN-extension HNN(G; A, B, Φ) as deﬁned in Section 7.3 and, as before, let X and Y be transversals for A and B in G with X ∩ Y = {1}. Consider the subset P = G ∪ GtY ∪ Gt−1 X ⊂ H. We deﬁne a partial multiplication by the obvious rules (left to the reader) according to the following table. G × G −→ G G × GtY G × Gt−1 X

−→ GtY −→ Gt−1 X

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83

GtY × G −→ GtY Gt−1 X × G −→ Gt−1 X Gt−1 X × GtY GtY × Gt

−1

X

−→ G if the inner part XG is in A −→ G if the inner part Y G is in B.

This deﬁnes a pregroup P for H, where D = G × G ∪ G × GtY ∪ G × Gt−1 X ∪ GtY × G ∪ Gt−1 X × G ∪ S, where S is the subset of Gt−1 X ×GtY ∪GtY ×Gt−1 X where inner parts XG or Y G belong to A or B, as appropriate. The partial multiplication table can be directly read from the convergent system we used in Section 7.3. As we shall see below, it deﬁnes an (inﬁnite) geodesically perfect rewriting system, where again we view elements of P as letters. Note also that we could replace X and Y by X = Y = G throughout the deﬁnition of our pregroup P in which the multiplication table could be slightly more simply described, but would be unnecessarily large. In [49] an alternative pregroup for H is deﬁned with underlying set consisting of equivalence classes of elements of G ∪ t−1 G ∪ Gt ∪ t−1 Gt under the equivalence relation generated by t−1 at ∼ Φ(a), for a ∈ A. However we feel that the resulting rewriting rules are obscured by the equivalence relation on the underlying set. The following is the principal result on the universal groups of pregroups. Theorem 8.3 (Stallings [49]). Let P be a pregroup. Then the following hold. 1) Every element of U (P ) can be represented by a P -reduced sequence; 2) any two P -reduced sequences representing the same element are ≈ equivalent, in particular they have the same length, and 3) P embeds into U (P ). 8.1. Rewriting systems for universal groups The result of [49] cited above may be regarded as showing that composition of the inclusion map P → P ∗ with the standard quotient map P ∗ → U (P ) is injective, where P ∗ is the free monoid on P . We show here how to achieve this with the help of a geodesically perfect Thue system. Since this approach may be new we work out the details. It is convenient to work over P ∗ and view each element of P as a letter. We have to distinguish whether a product is taken in the free monoid P ∗ or in P , and we introduce the following convention. Whenever we write [ab] we mean that (a, b) ∈ D ⊆ P × P with m(a, b) = [ab] ∈ P : that is the product ab is deﬁned in P and yields a letter. The system S = S(P ) ⊆ P ∗ × P ∗ is now deﬁned by the following rules. ε −→ 1 (= the empty word) ab −→ [ab] if (a, b) ∈ D ab ←→ [ac][c−1 b] if (a, c), (c−1 , b) ∈ D

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Theorem 8.4. Let P be a pregroup. Then the following hold. 1) P ∗ /S(P ) U (P ). 2) S is a geodesically perfect Thue system. Proof. Obviously, P ∗ /S deﬁnes U (P ) which proves 1). To prove 2) we show ﬁrst that the system S is strongly conﬂuent. For this we have to consider two rules such that the left-hand sides overlap. Strong conﬂuence involving only symmetric rules is trivial. Thus, we may assume that one rule is length-reducing. If one of the rules is ε −→ 1, then (by symmetry) the other rule is either εb −→ b or εb −→ c[c−1 b]. Since (c−1 , b) ∈ D implies (c, c−1 b) ∈ D and [c(c−1 b)] = b [49], both situations lead to b in at most one step. The next situation is: [ab]⇐=ab=⇒[ac][c−1 b] S

S

−1

Since (a, b) and (c , b) both belong to D we have (a, c(c−1 b)) ∈ D, as above, and (P4) implies that (ac, c−1 b) ∈ D, so we can apply the rule [ac][c−1 b] −→ [ab]. Finally, we have to consider: yd⇐=abd=⇒az S

S

∗

with a, b, d ∈ P and y, z ∈ P . We may assume that one rule is length-reducing of type ab −→ y = [ab]. The other rule is either of type bd −→ [bd] or of type bd ←→ [bc][c−1 d]. Assume ﬁrst that (b, d) ∈ D, then in both cases we can use: [ab]d=⇒[abb−1 ][bd] = a[bd] ⇐=a[bc][c−1 d]. S

S

The remaining case is that (b, d) ∈ / D and the situation is: [ab]d⇐=abd=⇒a[bc][c−1 d]. S

S

−1

Since (a, b), (b, c) and (c, c d) are in D, (P5) implies that either abc or bcc−1 d = bd is deﬁned in P . But bd is not deﬁned, therefore abc is deﬁned. We obtain: [ab]d=⇒[abc][c−1 d]⇐=a[bc][c−1 d]. S

S

Now we show that S is geodesic, from which it follows that it is geodesically perfect. Start with a sequence w ∈ P ∗ and apply only length-reducing rules until this is no longer possible. Clearly, the resulting sequence is P -reduced: ∗ w =⇒ a1 · · · an ∈ Γ∗ such that ai ai+1 is not deﬁned in P for 1 ≤ i < n. Possibly, S

one can still apply the symmetric rules, but we claim that any application of the symmetric rules gives again a P -reduced system. Indeed, assume u ∈ Γ∗ is P reduced, but it is not P -reduced after one application of a length-preserving rule from S(P ). Then there are four consecutive elements abde in u and an element c ∈ P , such that neither ab nor bd nor de is deﬁned, but bc, c−1 d are deﬁned and either a(bc) or (bc)(c−1 d) or (c−1 d)e is deﬁned. Assume the product a(bc) is deﬁned. Then the sequence a, bc, c−1 , d satisﬁes the premise of the axiom (P5), so either a(bc)c−1 = ab or (bc)c−1 d = bd must be deﬁned, contradicting the assumption that u is P -reduced. Similarly, (c−1 d)e cannot be deﬁned. Suppose now that

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(bc)(c−1 d) is deﬁned. Then the sequence b, c, c−1d satisﬁes the premise of (P4), since (bc) and c(c−1 d) are deﬁned. Since (bc)(c−1 d) is deﬁned (P4) implies that b(c−1 (cd)) = b(1d) = bd is deﬁned, in contradiction with P -reducibility of u. Remark 8.5. Stallings’ normal form Theorem 8.3 is now a consequence of Theorem 8.4 because elements from P are irreducible and the rewriting system is geodesically perfect. Thus, P -reduced sequences that deﬁne the same elements in U (P ) are ≈ equivalent. Remark 8.6. As above let Γ = P \ {ε}. Since S(P ) ⊆ P ∗ × P ∗ is strongly conﬂuent and geodesic, we obtain a geodesically perfect presentation of the universal group U (P ). In some sense it is however nicer to have such a presentation over Γ. So, let us put S (P ) ⊆ Γ∗ × Γ∗ deﬁned by the following rules: aa−1 ab ab

−→ 1 if a ∈ Γ −→ c if (a, b) ∈ D, a = b−1 , [ab] = c −1 ←→ [ac][c b] if (a, c), (c−1 , b) ∈ D

The diﬀerence is that a rule aa−1 −→ ε ∈ S (ε ∈ P is a letter) is replaced by aa−1 −→ 1 ∈ S (P ). This rule of S (P ) needs two steps of S(P ), but in S(P ) we win strong conﬂuence, whereas S (P ) is not strongly conﬂuent. However conﬂuence of S(P ) transfers to S (P ). Hence, both systems S(P ) and S (P ) are geodesically perfect. Using the geodesically perfect system S(P ) for U (P ) where P is ﬁnite we see that the result of Rimlinger [44] leads to the following statement which is slightly stronger than the result of [24]. Corollary 8.7. Let G be a ﬁnitely generated group. The following conditions are equivalent. 1) G is virtually free. 2) G can be presented by some ﬁnite geodesically perfect system. 3) G can be presented by some ﬁnite geodesic system. Proof. By Rimlinger [44], a ﬁnitely generated virtually free group is the universal group U (P ) of some ﬁnite pregroup P . By Theorem 8.4 it has a presentation by the (ﬁnite) geodesically perfect system S(P ). In our setting every geodesically perfect system is geodesic, so we get the implication from 2) to 3) for free. In order to pass from 3) to 1) one has to show that the set of words which are equivalent to 1 ∈ G forms a context-free language. This is can be demonstrated using an argument from [15], which has also been used in [24]. Consider a word w and write it as as w = uv such that u is geodesic. The preﬁx u is kept on a push down stack. Suppose that v = av , for some letter a. Push a onto the top of the stack: so the stack becomes ua. There is no reason to suppose that ua is geodesic and we perform length reducing reduction steps on it to produce an equivalent geodesic word u . Suppose this requires k steps: k

ua =⇒ u SR

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Let us show that we can bound k by some constant depending only on S. Indeed ∗ for all letters a we may ﬁx a word wa such that awa =⇒ 1. But this means SR

∗

u wa =⇒ u, SR

where u is geodesic and u represents the same group element as u did. But u was geodesic too. Hence |u| = |u|. Therefore | u| ≥ |u| − |wa | and this tells us k ≤ |wa |. Since k is bounded by some constant we see that the whole reduction process involves a bounded suﬃx of the word ua, only. This means we can factorise ua = pq and u = pr, where the length of q is bounded by some constant depending on k S only. Moreover, q =⇒ r. Since the length of q is bounded this reduction can be SR

performed using the ﬁnite control of the pushdown automaton. The automaton stops once the input has been read and then the stack gives us a geodesic corresponding to the input word w. In particular, the set of words which represent 1 in the group is context-free. Thus, the group presented is context-free; and using a result of Muller and Schupp [37] we see that G is virtually free. 8.2. Characterisation of pregroups in terms of geodesic systems In this section we consider Thue systems S ⊆ Γ∗ × Γ∗ corresponding to group presentations, i.e., Γ = X ∪ X −1 and S contains all the rules xx−1 → 1, x−1 x → 1, x ∈ X. We shall refer to these as group rewriting systems. We say that a rewriting system S ⊆ Γ∗ × Γ∗ is triangular if each rule → r ∈ S satisﬁes the “triangular” condition: || = 2, |r| ≤ 1, so every rule in S is of the form ab → c where a, b ∈ Γ and c ∈ Γ ∪ {1}. Observe that a triangular system is length-reducing. We also say that S is almost triangular if S = S ∪ S ◦ , where S is triangular and all rules in S ◦ are trivial, i.e., of the form a → 1, for some a ∈ Γ. Nontrivial examples of triangular systems come from triangulated presentations of groups. Namely, if X | R is a presentation of a group then one can triangulate this presentation by adding new generators and replacing old relations by ﬁnitely many triangular ones. Another type of example arises from pregroups. Let P be a pregroup. In Section 8.1 we deﬁned two rewriting systems S(P ) and S (P ) associated with P that deﬁne the universal group U (P ). Notice that the length-reducing part S (P )R of S (P ) is triangular (here Γ = P \ {ε}): S (P )R = {aa−1 → 1, ab → c | a, b, c ∈ Γ, (a, b) ∈ D, [ab] = c, a = b−1 }, meanwhile, the length reducing part S(P )R of S(P ) is almost triangular, since it contains the trivial rule ε → 1. Theorem 8.4 implies the following result. Corollary 8.8. Let P be a pregroup. Then S (P )R is a triangular geodesic system, S(P ) is an almost triangular geodesic system and U (P ) = Γ∗ /S (P )R = P ∗ /S(P )R .

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Proof. It suﬃces to observe that S (P )R and S (P ) deﬁne the same equivalence relation on Γ∗ . Indeed, every rule of the type ab → [ac][c−1 b], where [ac] and [c−1 b] are deﬁned, can be realized as the following rewriting sequence in S (P )R : ab ← acc−1 b → [ac]c−1 b → [ac][c−1 b], which shows that S (P )R and S (P ) are equivalent. The rest follows from Theorem 8.4 and Remark 8.6. To prove the converse of this corollary we need some notation. Let S ⊆ Γ∗ ×Γ∗ ∗ be a triangular group rewriting system, where Γ = X ∪ X −1. The congruence ⇐⇒ S

on Γ∗ induces an equivalence relation on the subset Γ ∪ {1}, which we denote by ≈. Deﬁne PS to be the quotient (Γ ∪ {1})/ ≈ and write [z] for the equivalence class of the element z ∈ Γ ∪ {1} and in addition ε for the equivalence class of 1. Deﬁne an involution p → p−1 on PS by setting [x]−1 = [x−1 ] and [x−1 ]−1 = [x], for x ∈ X, and setting ε−1 = ε. (Note that, since S is a group rewriting system, x ≈ 1 if and only if x−1 ≈ 1, so this involution is well deﬁned.) Now we deﬁne a “partial multiplication” on PS as follows. • For p, q ∈ PS \ {ε} the product pq is deﬁned and equal to s if there exist x, y ∈ Γ such that p = [x], q = [y] and there is a rule xy → z ∈ S, with z ∈ Γ ∪ {1} and s = [z]. • For all p ∈ PS we put pε = εp = p and pp−1 = p−1 p = ε. It is not hard to see that the partial multiplication on PS is well deﬁned. Lemma 8.9. Let S be a geodesic triangular group rewriting system. Then the following hold. 1) PS is a pregroup. 2) U (PS ) is isomorphic to the group Γ∗ /S. Proof. Clearly, the axioms P1) and P2) hold in PS by construction. It suﬃces to show that P4) and P5) hold in PS , in which case P3) follows. P4). If any one of p, q, r = ε then P4) holds trivially, so we may assume that p, q, r ∈ PS \ {ε}. Suppose then p = [a], q = [b], r = [c] ∈ PS and the products pq and qr are deﬁned, i.e., S contains rules ab → x and bc → y for some x, y ∈ Γ∪{1}. Suppose also that (pq)r is deﬁned in PS , so either [x] = [z] and zc → t ∈ S for some z, t ∈ Γ ∪ {1}, or [x] = ε, in which case let us deﬁne t = c. This means ∗ ∗ that abc ⇐⇒ t, for some t ∈ Γ ∪ {1} and also abc ⇐⇒ yc. As S is geodesic either S S

S

contains a rule yc → u, for some u ∈ Γ ∪ {1}, or y = 1, in which case let us deﬁne u = c. Then (pq)r = [t] = [u] = p(qr) in PS . It follows, by symmetry, that P4) holds. P5). Again we may assume we have p, q, r, s ∈ PS \ {ε} such that p = [a], q = [b], r = [c], s = [d] and the products pq, qr, rs are deﬁned; so there are rules ab → x, bc → y, cd → z ∈ S. We need to show that either pqr or qrs is deﬁned. Assume pqr is not deﬁned. This means in particular that y = 1 and that S contains no rule with left-hand side ay.

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We may rewrite abcd in two diﬀerent ways: abcd → xcd → xz and abcd → ayd. As S is geodesic either S must contain a rule which can be applied to ayd or one of a, y, d must be 1. Given our assumptions this means that S contains a rule with left-hand side yd. Thus we have (qr)s deﬁned, so P5) holds. This proves the ﬁrst statement. The second statement follows from Theorem 8.4, Remark 8.6 and Corollary 8.8. Indeed, it suﬃces to note that, by construction, the system S is the length reducing part of the system S (PS ) associated with the pregroup PS . Combining Corollary 8.8 and Lemma 8.9 one gets the following characterisation of pregroups and their universal groups in terms of triangular geodesic systems. Theorem 8.10. Let P be a pregroup. Then the reduced part of the rewriting system S (P ) is a geodesic triangular group system which deﬁnes the universal group U (P ). Conversely, if S is a triangular geodesic group system then PS is a pregroup, whose universal group is that deﬁned by S. This result gives a method of constructing a potentially useful pregroup for a group given by a presentation in generators and relators. It would be helpful to have a KB like procedure for ﬁnding such pregroups. Problem 8.11. Design an (KB-like) algorithm that for a given ﬁnite triangular rewriting system ﬁnds an equivalent triangular geodesic system.

References [1] P. Abramenko and K.S. Brown. Buildings, volume 248 of Graduate Texts in Mathematics. Springer, New York, 2008. Theory and applications. [2] J. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, and H. Short. Notes on word hyperbolic groups. Group Theory from a geometric viewpoint. World Scientiﬁc, Singapore, 1990. [3] G. Arzhantseva. An algorithm detecting Dehn presentations. Preprint, 2000. [4] A. Bj¨ orner and F. Brenti. Combinatorics of Coxeter groups, volume 231 of Graduate Texts in Mathematics. Springer, New York, 2005. [5] L.A. Bokut and L.-S. Shiao. Gr¨obner-Shirshov bases for Coxeter groups. Comm. Algebra, 29(9):4305–4319, 2001. Special issue dedicated to Alexei Ivanovich Kostrikin. [6] R. Book and F. Otto. String-rewriting systems. Texts and monographs in computer science. Springer-Verlag, 1993. [7] R.V. Book. Conﬂuent and other types of Thue systems. Journal of the Association for Computing Machinery, 29(1):171–182, 1982. [8] M.A. Borges-Trenard and H. P´erez-Ros´es. Complete presentations of Coxeter groups. Appl. Math. E-Notes, 4:1–6 (electronic), 2004. [9] N. Bourbaki. Lie groups and Lie algebras. Chapters 4–6. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley.

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[27] S. Hermiller and J. Meier. Algorithms and geometry for graph products of groups. J. Algebra, 171(1):230–257, 1995. [28] S.M. Hermiller. Rewriting systems for Coxeter groups. J. Pure Appl. Algebra, 92(2):137–148, 1994. [29] A.H.M. Hoare. Pregroups and length functions. Math. Proc. Cambridge Philos. Soc., 104(1):21–30, 1988. [30] D.F. Holt, B. Eick, and E.A. O’Brien. Handbook of computational group theory. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2005. [31] M. Jantzen. Conﬂuent String Rewriting, volume 14 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1988. [32] M. Kambites and F. Otto. Church-Rosser groups and growing context-sensitive groups. Journal of Automata, Languages and Combinatorics, 2008. To appear. [33] D.E. Knuth and P.B. Bendix. Simple word problems in universal algebras. In Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), pages 263–297. Pergamon, Oxford, 1970. [34] P. le Chenadec. Canonical Forms in Finitely Presented Algebras. Research Notes in Theoretical Computer Science. Pitman Publishing, Ltd., London-Boston, Mass, 1986. [35] R. Lyndon and P. Schupp. Combinatorial Group Theory. Classics in Mathematics. Springer, 2001. [36] W. Magnus, A. Karrass, and D. Solitar. Combinatorial Group Theory. SpringerVerlag, 1977. [37] D.E. Muller and P.E. Schupp. Groups, the theory of ends, and context-free languages. Journal of Computer and System Sciences, 26:295–310, 1983. [38] P. Narendran and R. McNaughton. The undecidability of the preperfectness of Thue systems. Theoret. Comput. Sci., 31(1-2):165–174, 1984. [39] P. Narendran and F. Otto. Preperfectness is undecidable for Thue systems containing only length-reducing rules and a single commutation rule. Information Processing Letters, 29:125–130, 1988. [40] M.H.A. Newman. On theories with a combinatorial deﬁnition of “equivalence.”. Ann. of Math. (2), 43:223–243, 1942. [41] M. Nivat and M. Benois. Congruences parfaites et quasi-parfaites. Technical Report 25e Ann´ee, Seminaire Dubreil, Paris, 1971/72. ´ unlaing. Undecidable questions related to Church-Rosser Thue systems. The[42] C. O’D´ oret. Comput. Sci., 23(3):339–345, 1983. [43] F. Otto and Y. Kobayashi. Properties of monoids that are presented by ﬁnite convergent string-rewriting systems – A survey. In Advances in Algorithms, Languages, and Complexity, pages 225–266, 1997. [44] F. Rimlinger. A subgroup theorem for pregroups. In Combinatorial group theory and topology (Alta, Utah, 1984), volume 111 of Ann. of Math. Stud., pages 163–174. Princeton Univ. Press, Princeton, NJ, 1987. ˇ 3:292–296, [45] A.I. Shirshov. Some algorithm problems for Lie algebras. Sibirsk. Mat. Z., 1962.

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[46] C.C. Sims. Computation with ﬁnitely presented groups. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1994. [47] C. Squier. Word problems and a homological ﬁniteness condition for monoids. J. of Pure and Applied Algebra, 49:201–217, 1987. [48] C. Squier, F. Otto, and Y. Kobayashi. A ﬁniteness condition for rewriting systems. Theoretical Computer Science, 131, 1994. [49] J. Stallings. Group theory and three-dimensional manifolds. Yale University Press, New Haven, Conn., 1971. A James K. Whittemore Lecture in Mathematics given at Yale University, 1969, Yale Mathematical Monographs, 4. [50] J.R. Stallings. Adian groups and pregroups. In Essays in group theory, volume 8 of Math. Sci. Res. Inst. Publ., pages 321–342. Springer, New York, 1987. [51] J. Tits. Le probl`eme des mots dans les groupes de Coxeter. In Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1, pages 175–185. Academic Press, London, 1969. [52] L. VanWyk. Graph groups are biautomatic. J. Pure Appl. Algebra, 94(3):341–352, 1994. [53] C. Wrathall. The word problem for free partially commutative groups. Journal of Symbolic Computation, 6(1):99–104, 1988. Volker Diekert Universit¨ at Stuttgart Universit¨ atsstr. 38 D-70569 Stuttgart, Germany Andrew J. Duncan Newcastle University Newcastle upon Tyne NE1 7RU, United Kingdom Alexei G. Myasnikov McGill University Montreal, Canada, H3A 2K6

Combinatorial and Geometric Group Theory Trends in Mathematics, 93–118 c 2010 Springer Basel AG

Regular Sets and Counting in Free Groups Elizaveta Frenkel, Alexei G. Myasnikov and Vladimir N. Remeslennikov Abstract. In this paper we study asymptotic behavior of regular subsets in a free group F of ﬁnite rank, compare their sizes at inﬁnity, and develop techniques to compute the probabilities of sets relative to distributions on F that come naturally from random walks on the Cayley graph of F . We apply these techniques to study cosets, double cosets, and Schreier representatives of ﬁnitely generated subgroups of F with an eye on complexity of algorithmic problems in free products with amalgamation and HNN extensions of groups. Mathematics Subject Classiﬁcation (2000). 20E05. Keywords. Geometric group theory, regular set, measures on free groups, Schreier transversals, generic and negligible sets.

1. Introduction During the last decade it has been realized that a natural set of algebraic objects usually can be divided into two parts. The large one (the regular part) consists of typical, “generic” objects; and the smaller one (the “singular” part) is made of “exceptions”. Essentially, this idea appeared ﬁrst in the form of zero-one laws in probability theory, number theory, and combinatorics. It became popular after seminal works of Erd˝ os, that shaped up the so-called Probabilistic Method (see, for example, [1]). In ﬁnite group theory the idea of genericity can be traced down to a series of papers by Erd˝ os and Turan in 1960-70’s (for recent results see, for example, [55]). In combinatorial group theory the concept of generic behavior is due to Gromov. His inspirational works [27, 28] turned the subject into an area of very active research, see for example, [2, 3, 4, 14, 7, 8, 13, 18, 15, 16, 17, 8, 9, 10, 32, 34, 35, 36, 37, 38, 39, 42, 50, 51, 53, 59, 61]. It turned out that the generic objects usually have much simpler structure, while the exceptions provide most of the diﬃculties. For instance, generic ﬁnitely generated groups are hyperbolic [27, 51], generic subgroups of hyperbolic groups are free [26], generic cyclically reduced elements in free groups are of minimal length in their automorphic orbits [39], generic automorphisms of a free group are strongly irreducible [52], etc.

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In practice, the generic-case analysis of algorithms is usually more important than the worst-case one. For example, knowing generic properties of objects one can often design simple algorithms that work very fast on generic elements. In cryptography, many successful attacks exploit the generic properties of random elements from a particular class, ignoring the existing hard instances [47, 48, 49, 54]. In the precise form the generic complexity of algorithmic problems appeared ﬁrst in the papers [34, 35, 14, 13]. We refer the reader to a comprehensive survey [25] on generic complexity of algorithms. In this paper we lay down some techniques that allow one to measure sets which appear naturally when computing with inﬁnite ﬁnitely presented groups. Our main idea is to approximate a given set by some regular subsets and estimate the asymptotic sizes of the regular sets using powerful tools of random walks on graphs and generating functions. The particular applications we have in mind concern with the generic complexity of the Word and Conjugacy problems in free products with amalgamation and HNN extensions. In general, such problems can be extremely hard. In [45] Miller described a free product of free groups with ﬁnitely generated amalgamation where the Conjugacy problem is undecidable; while in [45] he gave similar examples in the class of generalized HNN-extension of free groups. However, it has been proven in [15, 16] that on a precisely described set RP of “regular elements” in amalgamated free products and HNN extensions G the Conjugacy problem is decidable (under some natural conditions on the factors), furthermore, it is decidable in polynomial time. Namely, it was shown in [15, 16] that the group G (satisfying some natural assumptions) can be stratiﬁed into two parts with respect to the “hardness” of the conjugacy problem: • the Regular Part RP consists of so-called regular elements for which the conjugacy problem is decidable in polynomial time by the standard algorithms (described in [44, 43]). Moreover, one can decide whether or not a given element is regular in G; • the Black Hole BH (the complement of RP in G) consists of elements in G for which either the standard algorithms do not work at all, or they are slow, or the situation is not quite clear yet. The missing piece is to show that the set RP is, indeed, generic in G. This is not easy, the complete proof, which will appear in [24], relies on the techniques developed in the present paper. Now, a few words on the structure of the paper. In Section 2, following [14], we describe some techniques for measuring subsets in a free group F , the asymptotic classiﬁcation of large and small sets, and approximations via context-free and regular sets. In Section 3 we study, using graph techniques, Schreier system of representatives (transversal) of a ﬁnitely generated subgroup C in a free group F of ﬁnite rank. If S is a ﬁxed Schreier transversal of C then s ∈ S is called stable (on the right) if sc ∈ S for any c ∈ C. Intuitively, the stable representatives are “regular”, they are easy to deal with.

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In Section 4 we estimate the sizes of various subsets of F . In particular, we show that S is regular and thick (see deﬁnitions in Section 2), meanwhile the set Snst of non-stable representatives from S is exponentially negligible. Furthermore, the set Snst is exponentially negligible even relative to the set S. Our approach here is to “approximate” the sets in question by regular sets and to measure sizes of the regular sets using tools of random walks on graphs and Perron-Frobenius techniques. In Section 5 we develop a technique to compare sizes of diﬀerent regular sets at “inﬁnity” and give an asymptotic classiﬁcation of regular subsets of F relative to a ﬁxed preﬁx-closed regular subset L ⊆ F . The main result describes when regular subsets of L are “large” or “small” at inﬁnity in comparison to L. Notice, in the case when L = F , this result has been proven in [14] (Theorem 3.2).

2. Preliminaries In this section, following [14], we describe some techniques for measuring subsets in a free group F , the asymptotic classiﬁcation of large and small sets, and approximations via context-free and regular sets. 2.1. Asymptotic densities Let F = F (X) be a free group with basis X = {x1 , . . . , xm }. We use this notation throughout the paper. Let R be a subset of the free group F and Sk = { w ∈ F | |w| = k } the sphere of radius k in F . The fraction fk (R) =

|R ∩ Sk | |Sk |

is the frequency of elements from R among the words of length k in F. The asymptotic density ρ(R) of R is deﬁned by ρ(R) = lim sup fk (R). k→∞

R is called generic if ρ(R) = 1, and negligible if ρ(R) = 0. If, in addition, there exists a positive constant δ < 1 such that 1 − δ k < fk (R) < 1 for all suﬃciently large k then R is called exponentially generic. Meanwhile, if fk (R) < δ k for large enough k then R is exponentially negligible. In both the cases we refer to δ as a rate upper bound. Sometimes such sets are also called strongly generic or strongly negligible, but we refrain from this. The Cesaro limit 1 (1) ρc (R) = lim (f1 + · · · + fn ) . n→∞ n gives another asymptotic characteristic, called Cesaro density, or asymptotic average density. Sometimes, it is more sensitive then standard asymptotic density ρ

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(see, for example, [14], [60]). However, if limk→∞ fk (R) exists (hence is equal to ρ(R)) then ρc (R) also exists and ρc (R) = ρ(R). We will have to say more about ρc (R) below. Asymptotic density gives the ﬁrst coarse classiﬁcation of large (small) subsets: Coarse classiﬁcation 1) Generic sets; 2) visible or thick sets: the set R is visible if ρ(R) > 0; 3) negligible sets. Unfortunately, this classiﬁcation is very coarse, it does not distinguish many sets which, intuitively, have diﬀerent sizes. All our results in this paper concern with the strong version of the asymptotic density ρ, when the actual limit limk→∞ fk (R) exists. This allows one to diﬀerentiate sets with the same asymptotic density with respect to their growth rates. Thus generic sets R divide into subclasses of exponential, subexponential, superpolynomial, polynomial generic sets, with respect to the convergence rates of their frequency sequences {fk (R)}k∈N . The same holds for negligible sets as well. 2.2. Generating random elements and multiplicative measures One can use a no-return random walk Ws (s ∈ (0, 1]) on the Cayley graph C(F, X) of F with respect to the generating set X, as a random generator of elements of F (see [14]). We start at the identity element 1 and either do nothing with probability s (and return value 1 as the output of our random word generator), or move to one of the 2m adjacent vertices with equal probabilities (1 − s)/2m. If we are at a vertex v = 1, we either stop at v with probability s (and return the value v as the 1−s , to one of the 2m − 1 adjacent vertices output), or move, with probability 2m−1 lying away from 1, thus producing a new freely reduced word vx±1 i . Since the Cayley graph C(F, X) is a tree and we never return to the word we have already visited, it is easy to see that the probability μs (w) for our process to terminate at a word w is given by the formula μs (w) =

s(1 − s)|w| 2m · (2m − 1)|w|−1

for w = 1

(2)

and μs (1) = s. For R ⊆ F its measure μs (R) is deﬁned by μs (R) = lating μs (R) in terms of s, one gets μs (R) = s

∞

(3)

w∈R μs (w). Recalcu-

fk (1 − s)k ,

k=0

and the series on the right-hand side is convergent for all s ∈ (0, 1). The ensemble of distributions {μs } can be encoded in a single function μ(R) : s ∈ (0, 1) → μs (R) ∈ R.

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The argument above shows that for every subset R ⊆ F , μ(R) is an analytic function of s. It has been shown in [14] that μ(R) contains a lot of information about the asymptotic behaviour of the set R. To see where this information comes from renormalise the measures μs and consider the parametric family μ∗ = {μ∗s } of adjusted measures

1 2m · · μs (w). μ∗s (w) = (4) 2m − 1 s This new measure μ∗s is multiplicative in the sense that μ∗s (u ◦ v) = μ∗s (u)μ∗s (v),

(5)

where u◦v denotes the product of non-empty words u and v such that |uv| = |u|+|v| (no cancellation in the product uv). Moreover, if we denote t = μ∗s (x±1 i )=

1−s 2m − 1

(6)

then μ∗s (w) = t|w|

(7)

for every non-empty word w. Assume now, for the sake of minor technical convenience, that R does not contain the identity element 1. It is easy to see that μ∗s (R) =

∞

nk (R)tk

k=0

is the generating function of the spherical growth sequence nk (R) = |R ∩ Sk | of the set R in variable t which is convergent for each t ∈ [0, 1). √ The distribution μs has the uncomfortably big standard deviation σ = 1−s s , which reﬂects the fact that μs is strongly skewed towards “short” elements. The mean length of words in F distributed according to μs is equal to Ls = 1s − 1, so Ls → ∞ when s → 0. This shows that the asymptotic behaviour of the set R at “inﬁnity” (when Ls → ∞) depends on the behaviour of the function μ(R) when s → 0+ . Following [14], for a subset R of F we deﬁne a numerical characteristic μ0 (R) = lim+ μ(R) = lim+ s · s→0

s→0

∞

fk (1 − s)k .

k=0

If μ(R) can be expanded as a convergent power series in s at s = 0 (and hence in some neighborhood of s = 0): μ(R) = m0 + m1 s + m2 s2 + · · · , then μ0 (R) = lim μs (R) = m0 , s→0+

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and an easy corollary from a theorem by Hardy and Littlewood [30, Theorem 94] asserts that μ0 (R) is precisely the Cesaro limit ρc (R). A subset R ⊆ F is called smooth [14] if μ(R) can be expanded as a convergent power series in s at s = 0. 2.3. The frequency measure In this section we discuss the frequency measure, introduced in [14]. Let W0 be the no-return non-stop random walk on the Cayley graph C(F, X) of F (like Ws with s = 0), where the walker moves from a given vertex to any adjacent vertex away from the initial vertex 1 with equal probabilities 1/2m. In this event, the probability λ(w) that the walker hits an element w ∈ F in |w| steps (which is the same as the probability that the walker ever hits w) is equal to λ(w) =

1 , if w = 1, 2m(2m − 1)|w|−1

and λ(1) = 1.

This gives rise to a measure called the frequency measure on F , or Boltzmann distribution, deﬁned for subsets R ⊆ F by λ(R) = λ(w), w∈R

if the sum above is ﬁnite. One can view λ(R) as the cumulative frequency of R since ∞ λ(R) = fk (R). k=0

This measure is not probabilistic, since, for instance, λ(F ) = ∞, moreover, λ is additive, but not σ-additive. A subset R ⊆ F is called λ-measurable, or simply measurable (since we do not consider any other measures in this paper) if λ(R) < ∞. Every measurable set is negligible. Linear approximation. If the set R is smooth then the linear term in the expansion of μ(R) gives a linear approximation of μ(R): μ(R) = m0 + m1 s + O(s2 ). In this case, m0 = μ0 (R) is the Cesaro density of R. An easy corollary of [30, Theorem 94] shows that if μ0 (R) = 0 then m1 =

∞

fk (R) = λ(R).

k=1

On the other hand, even without assumption that R is smooth, if R is measurable, then μ(s) μ0 (R) = 0 and μ1 = lim = λ(R). s→0+ s

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2.4. Asymptotic classiﬁcation of subsets In this section we describe a classiﬁcation of subsets R in F , according to the asymptotic behavior of the functions μ(R). Recall, that the function μ(R) is analytic on (0, 1) for every subset R of F . R is smooth if μ(R) can be analytically extended to a neighborhood of 0. The subset R is called rational, algebraic, etc, with respect to μ if the function μ(R) is rational, algebraic, etc. Asymptotic classiﬁcation of sets. The following subtler classiﬁcation of sets in F (based on the linear approximation of μ(R)) was introduced in [14]: • Thick subsets: μ0 (R) exists, μ0 (R) > 0 and μ(R) = μ0 (R) + α0 (s), where

lim α0 (s) = 0.

s→0+

• Negligible subsets of intermediate density: μ0 (R) = 0 but μ1 (R) does not exist. • Sparse negligible subsets: μ0 (R) = 0, μ1 (R) exists and μ(R) = μ1 (R)s + α1 (s) where

lim

s→0+

α1 (s) = 0. s

• Exponentially negligible subsets. • Singular subsets. μ0 (R) does not exist. For sparse sets, the values of μ1 provide a further and more subtle discrimination by size. Lemma 2.1. [14] A subset is sparse in F if and only if it is measurable. 2.5. Context-free and regular languages as a measuring tool The simple observation in Section 2.2 that μ(R) is the generating function of the grows sequence {nk (R)}k∈N allows one to apply a well-established machinery of generating functions of regular and context-free languages to estimate asymptotic sizes of subsets R in F . We refer to [31] on regular and context-free languages, and to [21] on regular languages in groups. Algebraic sets and context free languages. If the set R is an (unambiguous) context free language then, by a classical theorem of Chomsky and Schutzenberger [19], the generating function μ∗ (R) = nk (R)tk , and hence the function μ(R), are algebraic functions of s. Moreover, if R is regular then μ(R) is a rational function with rational coeﬃcients [22, 57]. It is well known that singular points of an algebraic function are either poles or branching points. Since μ(R) is bounded for s ∈ (0, 1), this means that, for a context-free set R, the function μ(R) has no singularity at 0 or has a branching point at 0. A standard result on analytic functions allows us to expand μs (R) as a fractional power series: μs (R) = m0 + m1 s1/n + m2 s2/n + · · · ,

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n being the branching index. This technique was used in [16, 17] for numerical estimates of generic complexity of algorithms. If R is regular, than we actually have the usual power series expansion: μs (R) = m0 + m1 s + m2 s2 + · · · ; in particular, μ(R) can be analytically extended in the neighborhood of 0, so R is smooth. The following gives an asymptotic classiﬁcation of regular subsets of F . Theorem 2.2. [14, 6] 1) Every negligible regular subset of F is strongly negligible. 2) A regular subset of F is thick if and only if its preﬁx closure contains a cone. 3) Every regular subset of F is either thick or strongly negligible.

3. Schreier systems of representatives 3.1. Subgroup and coset graphs In this section for a given ﬁnitely generated subgroup of a free group we discuss its subgroup and coset graphs. Let F = F (X) be a free group with basis X = {x1 , . . . , xn }. We identify elements of F with reduced words in the alphabet X ∪ X −1 . Fix a subgroup C = h1 , . . . , hm of F generated by ﬁnitely many elements h1 , . . . , hm ∈ F . Following [33], we associate with C two graphs: the subgroup graph Γ = ΓC and the coset graph Γ∗ = Γ∗C . We freely use deﬁnitions and results from [33] in the rest of the paper. Recall, that Γ is a ﬁnite connected digraph with edges labeled by elements from X and a distinguished vertex (based-point) 1, satisfying the following two conditions. Firstly, Γ is folded, i.e., there are no two edges in Γ with the same label and having the same initial or terminal vertices. Secondly, Γ accepts precisely the reduced words in X ∪ X −1 that belong to C. To explain the latter observe, that walking along a path p in Γ one can read a word (p) in the alphabet X ∪ X −1 , the label of p (reading x in passing an edge e with label x along the orientation of e, and reading x−1 in the opposite direction). We say that Γ accepts a word w if w = (p) for some closed path p that starts at 1 and has no backtracking. One can describe Γ as a deterministic ﬁnite state automata with 1 as the unique starting and accepting state. For example, the graph Γ for the subgroup generated by x1 x2 x−1 is shown 1 in Figure 1 below. x 1 -r x2 r M 1 Figure 1

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Given the generators h1 , . . . , hm of the subgroup C, as words from F (X), one can eﬀectively construct the graph Γ in time O(n log∗ n) [58]. The coset graph (also known as the Schreier graph) Γ∗ = Γ∗C of C is a connected labeled digraph with the set {Cu | u ∈ F } of the right cosets of C in F as the vertex set, and such that there is an edge from Cu to Cv with a label x ∈ X if and only if Cux = Cv. One can describe the coset graph Γ ∗ as obtained from Γ by the following procedure. Let v ∈ Γ and x ∈ X such that there is no outgoing or incoming edge at v labeled by x. We call such v a boundary vertex of Γ and denote the set of such vertices by ∂Γ. For every such v ∈ ∂Γ and x ∈ X we attach to v a new edge e (correspondingly, either outgoing or incoming) labeled x with a new terminal vertex u (not in Γ). Such vertices u are called frontier vertices, we denote the set of frontier vertices of Γ by ∂ + Γ. Then we attach to u the Cayley graph C(F, X) of F relative to X (identifying u with the vertex 1 of C(F, X)), and then we fold the edge e with the corresponding edge in C(F, X) (that is labeled x and is incoming to u). Observe, that for every vertex v ∈ Γ∗ and every reduced word w in X ∪ X −1 there is a unique path Γ∗ that starts at v and has the label w. By pw we denote such a path that starts at !1, and by" vw the end vertex of pw . Here is the fragment of the graph Γ∗ for C = x1 x2 x−1 : 1 6 -

6

-

x2 6 -r 6

6 x1 - r1 6

x1 1 r - r ix6

x2

x2

-r 6 Figure 2 Lemma 3.1. Γ∗C is the coset graph of C in F . Proof. See, for example, [33]. ∗

Notice that Γ = Γ if and only if the subgroup C has ﬁnite index in F . Indeed, Γ = Γ∗ if and only if for every vertex v of Γ and every label x ∈ X, there is an edge in Γ labeled by x which exits from v, and an edge with label x which enters v, but this is precisely the characterization of subgroups of ﬁnite index in F [33, Proposition 8.3].

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A spanning subtree T of Γ with the root at the vertex 1 is called geodesic if for every vertex v ∈ V (Γ) the unique path in T from 1 to v is a geodesic path in Γ. For a given graph Γ one can eﬀectively construct a geodesic spanning subtree T (see, for example, [33]). From now on we ﬁx an arbitrary spanning subtree T of Γ. It is easy to see that the tree T uniquely extends to a spanning subtree T ∗ of Γ∗ . Let V (Γ∗ ) be the set of vertices of Γ∗ . For a subset Y ⊆ V (Γ∗ ) and a subgraph Δ of Γ∗ , we deﬁne the language accepted by Δ and Y as the set L(Δ, Y, 1) of the labels (p) of paths p in Δ that start at 1 and end at one of the vertices in Y , and have no backtracking. Notice that the words (p) are reduced since the graph Γ∗ is folded. Notice, that F = L(Γ∗ , V (Γ∗ ), 1) and C = L(Γ, {1}, 1) = L(Γ∗ , {1}, 1). Sometimes we will refer to a set of right (left) representatives of C as the right (left) transversal of C. Furthermore, to simplify terminology, a transversal will usually mean a right transversal, if not said otherwise. Recall, that a transversal S of C is termed Schreier if every initial segment of a representative from S belongs to S. Proposition 3.2. Let C be a ﬁnitely generated subgroup of F . Then: 1) for every spanning subtree T of Γ the set ST ∗ = L(T ∗ , V (Γ∗ ), 1) is a Schreier transversal of C. 2) For every Schreier transversal S of C there exists a spanning subtree T of Γ such that S = ST ∗ . Proof. The statement 1) follows directly from Lemma 3.1. To prove 2) notice that every reduced path p ∈ Γ∗ can be decomposed as p = pint ◦ pout , where pint is a maximal reduced path in Γ, and pout is the tail of p outside of Γ. This decomposition is unique. Moreover: • if v ∈ V (Γ) and p is a reduced path from 1 to v in Γ∗ then p passes only through vertices of Γ; • if v ∈ V (Γ∗ ) \ V (Γ) and p and p are two paths from 1 to v, where p = pint ◦ pout , and p = pint ◦ pout , then pout = pout . Let S be a Schreier transversal of C in F and s ∈ S. Suppose that the reduced path ps ends at some vertex vs in Γ. Then the whole path ps lies in Γ. Let T be a subgraph of Γ generated by the union of all paths ps , where s ∈ S and vs ∈ Γ. Since S is a Schreier transversal, T is a maximal subtree of Γ. It is clear that S = ST ∗ . Hence, the result. Proposition 3.2 allows one to identify elements from a given Schreier transversal S of C with the vertices of the graph Γ∗ , provided a maximal subtree of Γ is ﬁxed. We use this frequently in the sequel. Corollary 3.3. The number of distinct Schreier transversals of C in F is ﬁnite and equal to the number of spanning subtrees of Γ.

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3.2. Schreier transversals In this section we introduce various types of representatives of C in F relative to a ﬁx basis X of F . Deﬁnition 3.4. Let S be a transversal of C. • A representative s ∈ S is called internal if the path ps ends in Γ, i.e., vs ∈ V (Γ). By Sint we denote the set of all internal representatives in S. Elements from Sext = S Sint are called the external representatives in S. • A representative s ∈ S is called geodesic if it has minimal possible length in its coset Cs. The transversal S is geodesic if every s ∈ S is geodesic. • A representative s ∈ S is called singular if it belongs to the generalized normalizer of C: NF∗ (C) = {f ∈ F |f −1 Cf ∩ C = 1}. All other representatives from S are called regular. By Ssin and, respectively, Sreg we denote the sets of singular and regular representatives from S. • A representative s ∈ S is called stable (on the right) if sc ∈ S for any c ∈ C. By Sst we denote the set of all stable representatives in S, and Snst = S Sst is the set of all non-stable representatives from S. In the following proposition we collect some basic properties of various types of representatives. Recall that the cone deﬁned by (or based at) an element u ∈ F is the set C(u) of all reduced words in F that have u as an initial segment. In this case C(u) = {w ∈ F | w = u ◦ v, v ∈ F }. Proposition 3.5. Let S be a Schreier transversal for C, so S = ST ∗ for some spanning subtree T ∗ of Γ∗ . Then the following hold: 1) Sint is a basis of C, in particular, |Sint | = |V (Γ)|. 2) Sext is the union of ﬁnitely many coni C(u), where vu ∈ ∂ + Γ. 3) Ssin is contained in a ﬁnite union of double cosets Cs1 s−1 2 C of C, where s1 , s2 ∈ Sint . 4) Snst is a ﬁnite union of left cosets of C of the type s1 s−1 2 C, where s1 , s2 ∈ Sint . Proof. 1) is well known, see [12], for example. 2) follows immediately from the construction. To see 3) notice ﬁrst that Ssin ⊆ NF∗ (C) and NF∗ (C) is the union of ﬁnitely many double cosets CsC, where s ∈ Ssin , and furthermore, every such coset has the form CsC = Cs1 s−1 2 C, where s1 , s2 ∈ Sint (see Lemma 5 in [12], or Propositions 9.8 and 9.11 in [33], or Theorem 2 in [32]). To see 4) assume that s ∈ S is not stable, so there exists an element c ∈ C such that sc ∈ S. Then s = s1 ◦ t, c = t−1 ◦ d, sc = s1 ◦ d. We claim, that the terminal vertex of s1 lies in Γ (viewing s as a path in Γ∗ ). Indeed, if not, then s1 , as well as s1 ◦ d, is in Sext – contradiction. Hence, s1 ∈ Sint . Since c = t−1 ◦ d there is a closed path in Γ with the label t−1 ◦ d, starting at 1C . Let s2 ∈ Sint be the representative of t−1 . Then s2 d = c1 ∈ C, hence sc = s1 ◦ d = s1 s−1 2 c1 , so s ∈ s1 s−1 2 C, as claimed.

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Proposition 3.6. Let S be a Schreier transversal for C, so S = ST ∗ for some spanning subtree T ∗ of Γ∗ . Then the following hold: 1) If T ∗ is a geodesic subtree of Γ∗ (and hence T is a geodesic subtree of Γ) then S is a geodesic transversal. 2) If C is a malnormal subgroup of F then Ssin = ∅. 3) Ssin ⊆ Snst . Proof. 1) is straightforward (see also [33]). 2) If C is malnormal then NF∗ (C) = 1, so Ssin = ∅. 3) If s ∈ Ssin then c = s−1 c1 s for some non-trivial c, c1 ∈ C, so c1 s = sc. Since sc = s we conclude that sc ∈ S, hence s ∈ Snst .

4. Measuring subsets of F Recall that a ﬁnite automaton A is a ﬁnite labeled oriented graph (possibly with multiple edges and loops). We refer to its vertexes as states. Some of the states are called initial states, some accept or ﬁnal states. We assume further that every edge of the graph is labeled by one of the symbols x±1 , x ∈ X, where F = F (X) is a free group of ﬁnite rank m. The language accepted by an automaton A is the set L = L(A) of labels on paths from initial to accept states. An automaton is said to be deterministic if, for any state there is at most one arrow with a given label exiting the state. A regular set is a language accepted by a ﬁnite deterministic automaton. The following facts about regular sets are well known. Let A and B are regular subsets in F . Then: • the sets A ∪ B, A ∩ B and A B are regular. • The preﬁx closure A of a regular set A is regular. Here, the preﬁx closure A is the set of all initial segments of all words in A. • If ab = a ◦ b for any a ∈ A, b ∈ B then AB is regular. • If ab = a ◦ b for any a, b ∈ A then A∗ is regular. The following notation is useful. For u, v ∈ F deﬁne c(u, v) = 12 (|u| + |v| − |uv|), the amount of cancellation in the product uv. Proposition 4.1. Let R1 and R2 be subsets of F . Then the following statements hold: 1) If R1 ⊆ R2 and R2 is negligible (exponentially negligible) then so is R1 . 2) If R1 , R2 are negligible (exponentially negligible) then so is R1 ∪ R2 . 3) If R1 and R2 are negligible (exponentially negligible) then so is the set R1 ◦ R2 = {r1 r2 | ri ∈ Ri , c(r1 , r2 ) = 0}. 4) If R1 and R2 are negligible (exponentially negligible) then so is the set R1 ◦ R2 = {r1 r2 | ri ∈ Ri , c(r1 , r2 ) ≤ t}. t

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Proof. The proof is straightforward. Deﬁnition 4.2. Let R1 and R2 be subsets of F and f : R1 → R2 a map. Then:

• f is called d-isometry, where d is a non-negative real number, if for any w ∈ R1 |w| − d ≤ |f (w)| ≤ |w| + d. • f has uniformly bounded ﬁbers if there exists a constant c such that every element w ∈ R2 has at most c pre-images in R1 . Proposition 4.3. Let R1 and R2 be subsets of F . Then the following statements hold: 1) If f : R1 → R2 is a surjective d-isometry and R1 is negligible (exponentially negligible) then so is R2 . 2) If f : R1 → R2 is a d-isometry with uniformly bounded ﬁbers and R2 is negligible (exponentially negligible) then so is R1 . Proof. Notice that for k > d k+d

fk (R2 ) ≤

fj (R1 ),

j=k−d

and 1) follows. Similarly, fk (R1 ) ≤ c

k+d

fj (R2 )

j=k−d

for k > d and 2) follows.

Proposition 4.4. Let C be a ﬁnitely generated subgroup of inﬁnite index of a free group F . Then every Schreier transversal of C in F is regular and thick. Proof. By deﬁnition S = Sint ∪ Sext . By Proposition 3.5 the set Sint is ﬁnite and the set Sext is a non-empty ﬁnite union of cones. By Theorem 2.2 each cone in Sext is thick. Therefore, the set Sext , as well as the set S, is thick. Clearly, every cone is regular, so is the set S. Proposition 4.5. Let C be a ﬁnitely generated subgroup of inﬁnite index in F . Then the following hold: 1) C is exponentially negligible in F and one can ﬁnd some upper bound δ < 1 for the growth rate of C. 2) Every coset of C in F is exponentially negligible in F . Proof. 1) follows from Proposition 1 and Corollary 1 in [6]. 2) follows from 1) above and 4) from Proposition 4.1.

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Proposition 4.6. Let C be a ﬁnitely generated subgroup of inﬁnite index in F . Then the following hold: f 1) C ∗ = C is exponentially negligible in F . f ∈F

2) For every c ∈ C the set conjugacy class cF = {f −1cf |f ∈ F } is exponentially negligible in F . Proof. The statement 1) has been shown in [14] and also in Proposition 1.10 in [5]. The statement 2) is shown in Proposition 1.11 in [5]. Proposition 4.7. Let C be a ﬁnitely generated subgroup of inﬁnite index in F and S is a Schreier transversal of C in F . If S0 ⊆ S is a exponentially negligible subset of F , then the set Cs is exponentially negligible in F. s∈S0

Proof. By Proposition 3.5 S = Sint ∪ Sext , where Sint is a ﬁnite set and Sext is a union of ﬁnitely many cones C(u), where u ends at ∂ + Γ, i.e., vu ∈ ∂ + Γ. It suﬃces to prove the result for S0 ∩ C(u) for a ﬁxed vu ∈ ∂ + Γ. To this end we may assume from the beginning that S0 ⊆ C(u). If s is the representative of u in S, then every word from C(u) contains s as an initial segment. Since s is not readable in Γ the amount of cancellation c(w, t) in the product wt, where w ∈ C and t ∈ C(u) does not exceed the length of s. Hence CS0 = C ◦ S0 |s|

and the result follows from the statement 4) of Proposition 4.1.

Proposition 4.8. Let A and B be ﬁnitely generated subgroups of inﬁnite index in F . Then for any w ∈ F the double coset AwB is exponentially negligible in F. −1

Proof. Observe, that AwB = AB w w, so by the statement 2) of Proposition −1 −1 4.5 it suﬃces to show that AB w is exponentially negligible. Since B w is just another ﬁnitely generated subgroup of inﬁnite index in F one can assume from the beginning that w = 1. Let S be a geodesic Schreier transversal for A in F. Then As AB = s∈S0

for some subset S0 ⊆ S. By Proposition 4.7 it suﬃces to show that the subset S0 is exponentially negligible. Since the set Sint is ﬁnite we may assume that S0 ⊆ Sext . Now we construct an r-isometry α : S0 → B. Let TA be the spanning subtree of ΓA such that S = STA ∗ and TB be a spanning geodesic subtree of ΓB . Denote by d the maximum of the diameters of the trees TA and TB . To describe the map α choose an arbitrary element s ∈ S0 . Without loss of generality assume that |s| ≥ d, because there are only ﬁnitely many such s that have smaller length and by Proposition 4.5 they will not extremely change asymptotic size of AB since A of inﬁnite index in F. Then as = b for some a ∈ A and b ∈ B. We claim that there exists an element bs ∈ B such that |sb−1 s | ≤ 2d. Indeed, the cancellation in the product as is at most d (see the argument in Proposition 4.7). Hence s and b have a

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common terminal segment t of length at least |s| − d (recall that |s| ≥ d). It follows that in the graph ΓB there exists a path from some vertex v to 1B with the label tv . −1 −1 Then bs = tv t ∈ B and |sb−1 tv | ≤ 2d. Hence s and bs has a long common s | = |st terminal segment and diﬀer only on the initial segment of length at most 2d. It follows that the map α : s → bs gives a 2d-isometry α : S0 → B. Notice that α has uniformly bounded ﬁbers. Indeed, if α(s1 ) = α(s2 ) = b then s1 and s2 diﬀer from b, hence from each other, only on the initial segment of length at most 2d. So there are at most (2d)2|X| such distinct elements. Since B is exponentially negligible by Proposition 4.3 the set S0 is also exponentially negligible, as claimed. This proves the result. Notice, that the property being geodesic for Schreier transversal S for A in F is not crucial for our prove. Namely, for arbitrary Schreier transversal S all conclusions can be repeated with slightly diﬀerent constant. Now we can state the main result of the section. Theorem 4.9. Let C be a ﬁnitely generated subgroup of inﬁnite index in F and S a Schreier transversal for C. Then the following hold: 1) The generalized normalizer NF∗ (C) of C in F is exponentially negligible in F . 2) The set of singular representatives Ssin is exponentially negligible in F . 3) The set Snst of unstable representatives is exponentially negligible in F . Proof. To see 1) recall that the generalized normalizer NF∗ (C) of C in F is a ﬁnite union of double cosets of C in F. Therefore NF∗ (C) is exponentially negligible in F by Proposition 4.8. 2) follows immediately from 1). To prove 3) observe that Snst is a ﬁnite union of left cosets of C (see 3) in Proposition 3.5). Now the result follows from Proposition 4.5. Theorem 4.9 can be strengthen as follows. Corollary 4.10. Let C be a ﬁnitely generated subgroup of inﬁnite index in F. Then the sets Sin (C) = Ssin , Uns (C) = Snst , S

S

where S runs over all Schreier transversals of C, are exponentially negligible. Proof. By Corollary 3.3 there are only ﬁnitely many Schreier transversals of C. Now the result follows from Theorem 4.9 and Proposition 4.1.

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5. Comparing sets at inﬁnity 5.1. Comparing Schreier representatives In this section we give another version of Theorem 4.9. To explain we need a few deﬁnitions. For subsets R, L of F we deﬁne their size ratio at length k by fk (R) |R ∩ Sk | fk (R, L) = = . fk (L) |L ∩ Sk | The size ratio ρ(R, L) at inﬁnity of R relative to L (or the relative asymptotic density) is deﬁned by ρ(R, L) = lim sup fk (R, L). k→∞

By rL (R) we denote the cumulative size ratio of R relative to L: rL (R) =

∞

fk (R, L).

k=1

We say that R is L-measurable, if rL (R) is ﬁnite. R is called negligible relative to L if ρ(R, L) = 0. Obviously, an L-measurable set is L-negligible. A set R is termed exponentially negligible relative to L (or exponentially L-negligible) if fk (R, L) ≤ q k for all suﬃciently large k. The following result is simple, but useful. Proposition 5.1. Let R be an exponentially negligible set in F . 1) For any w ∈ F the set R is a exponentially negligible relative to the cone C(w). 2) The set R is exponentially negligible relative to any exponentially generic subset T of F . Proof. Observe, that fk (C(w)) = 1/2m(2m − 1)|w|−1 is a constant. Since fk (R, C(w)) =

fk (R) fk (C(w))

it follows that R is exponentially negligible relative to C(w). This proves 1). To prove 2) denote by p and q the corresponding rate bounds for R and T , so fk (R) ≤ pk , fk (T ) ≥ 1 − q k for suﬃciently large k. Then, for such k,

k fk (R) p pk = . fk (R, T ) = ≤ 1 fk (T ) 1 − qk (1 − q k ) k Since p lim 1 = p k→∞ (1 − q k ) k it follows that for any ε > 0 fk (R, T ) ≤ (p + ε)k for suﬃciently large k, as claimed.

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Corollary 5.2. Let C be a ﬁnitely generated subgroup of inﬁnite index in F and S a Schreier transversal for C. Then the following hold: 1) The set of singular representatives Ssin is exponentially negligible in S. 2) The set Snst of unstable representatives is exponentially negligible in S. Proof. The statements of this corollary follow immediately from Theorem 4.9 and Propositions 3.5, 3.6 and 5.1. 5.2. Comparing regular sets In this section we give an asymptotic classiﬁcation of regular subsets of F relative to a ﬁxed preﬁx-closed regular subset L ⊆ F . For this purpose we are going to describe how one can use a random walk on the ﬁnite automaton B recognizing regular subset R ⊆ L similar to the one in Section 2.3. It will be convenient to further put B to special form consistent to L. Recall Myhill-Nerode’s theorem on regular languages (see, for example, [21], Theorem 1.2.9). For a language R over an alphabet A consider an equivalence relation ∼ on A∗ deﬁned as follows: two strings w1 and w2 are equivalent if and only if for each string u over A the words w1 u and w2 u are either simultaneously in R or not in R. Then R is regular if and only if there are only ﬁnitely many ∼-equivalence classes. Now, let R ⊆ L. Deﬁne an equivalence relation ∼ on L such that w1 ∼ w2 if and only if for each u ∈ F the following condition holds: w1 u = w1 ◦ u and w1 u ∈ R if and only if w2 u = w2 ◦ u and w2 u ∈ R. The following is an analog of Myhill-Nerode’s theorem for free groups. Lemma 5.3. Let R ⊆ L ⊆ F and L preﬁx-closed and regular. Then R is regular if and only if there are only ﬁnitely many ∼-equivalence classes in L. Proof. The proof is similar to the original one. We give a short sketch of the most interesting part of it. If the set of the equivalence classes is ﬁnite one can deﬁne an automaton B on the set of equivalence classes as states. If x ∈ X ∪ X −1 and [w] is the equivalence class of some w such that w ◦ x ∈ R then one connects the state [w] with an edge labeled by x to the state [wx]. The class [ε], where ε is the empty word, is the initial state, while a state [w] is an accepting state if and only if w ∈ R. In this case L(B) = R. Since R is regular, we suppose that B as in Lemma 5.3 and modify it in the next way. Without loss of generality we can assume that A is in the normal form, i.e., it has only one initial state I and doesn’t contain inaccessible states. Let S = [w] be a state of B. Denote by S pr the uniquely deﬁned state in A which is the terminal state of the path with the label w in A, starting at [ε]. The state S pr is well deﬁned, it does not depend on the choice of w. We call S pr the prototype of S. Since B accepts only reduced words in X ∪ X −1 one can transform B to a form where the following hold:

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a) B has only one initial state I and one accepting state Z. b) For any state S of B, all arrows which enter S have the same label x ∈ X∪X −1 and arrows exiting from S cannot have label x−1 (this can be achieved by splitting the states of B, see Figure 1). We shall say in this situation that S has type x. c) For every state S of B there is a direct path from S to the accept state Z. d) There are no arrows entering the initial state I. m @ a m @ c * @ R @ m a - Am Hd HH j m b

m Ha HH j m c- m A a * @ c m @ d @ R @ - m Am d * b m

=⇒

m

Figure 3. Splitting the states of the automaton B. The ﬁnal version of obtained automaton B we will call an automaton consistent with A. Now we are ready to deﬁne a no-return random walk on B as it was claimed above. Namely, let B be consistent with A and let S be a state in B. Denote by ν = ν(S pr ) the number of edges exiting from the prototype state S pr in A. The walker moves from S along some outgoing edge with the uniform probability ν1 . In this event, the probability that the walker hits an element w ∈ R in |w| steps (when starting at [ε]) is the product of frequencies of arrows in a direct path from the initial state I to the accept state Z with the label w. It is not hard to see that this probability is equal to λL (w). This gives rise to the measure λL on R: λL (R) =

λL (w) =

w∈R

where fk (R, L) =

∞

fk (R, L),

k=0

λL (w).

w∈R∩Sk

Note that, generally speaking, fk (R, L) diﬀers from fk (R, L) deﬁned in Section 5.1. Indeed, walking in B we have diﬀerent number of possibilities to continue our walk on the next step depending on way we chose. On the other hand, fk (R, F (X)) = fk (R, F (X)). Now we can use the tools of random walks to compute λL (R). Notice, that λL is multiplicative, i.e., λL (uv) = λL (u)λL (v)

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for any u, v ∈ R such that uv = u ◦ v and uv ∈ R. We say that R is λL -measurable, if λL (R) is ﬁnite. A set R is termed exponentially λL -measurable, if fk (R, L) ≤ q k for all suﬃciently large k. The following result is simple, but useful. Let w ∈ F. The set CL (w) = L ∩ C(w) is called an L-cone. Obviously, CL (w) is a regular set. We say that CL (w) is L-small, if it is exponentially λL -measurable. The following is the main result of this section. Theorem 5.4. Let R be a regular subset of a preﬁx-closed regular set L in a free group F. Then either the preﬁx closure R of R in L contains a non-small L-cone or R is exponentially λL -measurable. Before proving the theorem we establish a few preliminary facts. We ﬁx a preﬁx-closed regular subset L of F . Proposition 5.5. Let R1 and R2 be subsets of F . Let also P be one of the properties { “to be L-measurable”, “to be exponentially L-negligible”, “to be λL -measurable”, “to be exponentially λL -measurable”}. Then the following hold: 1) If R1 ⊆ R2 and R2 has property P then so is R1 . 2) If R1 , R2 have property P then so is R1 ∪ R2 . 3) If R1 and R2 have property P then so is the set R1 ◦ R2 = {r1 r2 | ri ∈ Ri , c(r1 , r2 ) = 0}.

Proof. The proofs are easy.

To strengthen the last statement in Proposition 5.5 we need the following ◦ = Tk◦ ◦ T . notation. For a subset T ⊆ F put T1◦ = T and deﬁne recursively Tk+1 Denote ∞ ◦ T∞ = Tk◦ . k=1

Lemma 5.6. Let T be a regular set and a number q, 0 < q < 1, such that fk (T, L) ≤ ◦ is exponentially λL -measurable. q k for every positive integer k. Then the set T∞ ◦ of length k comes in the form w = u1 ◦ u2 ◦ · · · ◦ ut , Proof. Every word w ∈ T∞ where ui ’s are non-trivial elements from T and k = |u1 | + · · · + |ut |. On the other hand, if k = k1 + · · · + kt is an arbitrary partition of k into a sum of positive integers and u1 , . . . , ut are words in T such that |ui | = ki , then w = ◦ ◦ u 1 · · · u t ∈ T∞ . Since λL is multiplicative every partition of k adds to fk (T∞ , L) a ◦ ◦ k1 +···+kt number fk1 (T∞ , L) . . . fkt (T∞ , L), which is bounded from above by q = qk . If p(k) is the number of all partitions of k into a sum of positive integers then ◦ , L) ≤ p(k)q k . It is known (Hardy and Ramanujan) that fk (T∞ √ 2k eπ 3 √ . p(k) ∼ 4k 3 ◦ ◦ , L) < q1k , for some 0 < q < q1 < 1 and all suﬃciently large k, so T∞ Hence fk (T∞ is exponentially λL -measurable, as claimed.

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Proof of Theorem 5.4. In the most part we follow the proof of Theorem 2.2 from [14]. Suppose that all L-cones in R are non-small. Since R ⊆ R by Proposition 5.5 we can assume that R itself is preﬁx-closed in L. We have to prove that R is exponentially λL -measurable. Let R = L(B) and B consistent to A (where A recognize L). It is convenient to further split B into two parts. Denote by B1 the automaton obtained from B by removing all arrows exiting from Z.

Figure 4. The automaton B.

Figure 5. The automaton B1 .

Let B2 be the automaton formed by all states in B that are accessible from the state Z, with the same arrows between them as in B; Z is the only initial and accepting state of B2 .

Figure 6. The automaton B2 . We assign to arrows in B1 and B2 the same frequencies as to the corresponding arrows in B. If R1 and R2 are the languages accepted by B1 and B2 then, obviously, R = R1 ◦ R2 . By Proposition 5.5 to prove the theorem it suﬃces to show that R1 and R2 are exponentially λL -measurable.

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Claim. The set R2 is exponentially λL -measurable. Proof of the claim. Notice, that for every w ∈ R1 w ◦ R2 ⊆ L(A) = R and w ◦ R2 is an L-cone. It is easy to see, that R2 is exponentially λL -measurable if and only if so w ◦ R2 is. Let R3 ⊆ R2 be the subset consisting of those non-trivial words w ∈ R2 , whose paths pw visit the state Z of B2 only once. The set R3 is regular – it is accepted by an automaton B3 , which is obtained from B2 by splitting the state Z into two separate states: the initial state Z1 and an accepting state Z2 , in such a way that the edges exiting from Z in B2 are now exiting from Z1 and there no ingoing edges at Z1 , while there are no edges exiting from Z2 and all those arrows incoming in Z in B2 are now incoming into Z2 .

Figure 7. The automaton B3 . It follows immediately from the construction, that R2 = {ε} R3 (R3 ◦ R3 ) (R3 ◦ R3 ◦ R3 ) · · · = (R3 )◦∞ so λL (R2 ) = λL (R3 ) + (λL (R3 ))2 + (λL (R3 ))3 + · · · . (8) By Lemma 5.6 it suﬃces to show that there is a number q, 0 < q < 1, such that fk (R3 , L) ≤ q k for every k (not only for suﬃciently large k). It is not hard to see that this condition holds if R3 is exponentially λL -measurable and λL (R3 ) < 1, so it suﬃces to prove the latter two statements. By our assumption all L-cones in R = R are L-small. If for every state [w] = S in B2 and every x ∈ X ∪ X −1 there is an outgoing edge labeled by x at [w] if and only if the same holds for the state S pr in A (i.e., B2 is X-complete relative to A) then for every given w ∈ R1 one has C(w) ∩ R = w ◦ R2 = C(w) ∩ L, so w ◦ R2 is an L-cone. Hence it is L-small, i.e., exponentially λL -measurable, but then the set R2 , hence R2 , is exponentially λL -measurable, as claimed. This implies that for some state S there are less then ν = ν(S pr ) arrows exiting from S. Consider a ﬁnite Markov chain M with the same states as in B3

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together with an additional dead state D. We set transition probabilities from Z2 to Z2 and from D to D being equal 1. Every arrow from a state S in B3 gives the corresponding transition from the state S in M which we assign the transition 1 probability . If at some state S of type x in B3 there is no exiting arrow labeled ν y ∈ (X ∪X −1 ){x−1 }, in M we make a transition from S to D with the transition 1 probability . This describes M. ν

Figure 8. The automaton M. The states Z2 and D of Markov chain M are absorbing, and all other states are transient. The probability distribution on M concentrated at the initial state Z1 , converges to the steady state P which is zero everywhere with the exception of the two absorbing states Z2 and D. Obviously, P (Z2 ) = λL (R3 ). Since P (D) = 0 we have λL (R3 ) = P (Z2 ) < 1, so one of the required conditions on R3 holds (for more details on this proof we refer to [14, 41]). The other one follows directly from Corollary 3.1.2 in [41], which claims that in this case R3 is exponentially λL -measurable. This proves the claim. A similar argument shows that R1 is exponentially λL -measurable. This proves the theorem. Acknowledgement The authors thank Alexandre Borovik for very fruitful discussions.

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[59] A. Martino, T. Turner, and E. Ventura. The density of injective endomorphisms of a free group. Preprint, CRM, Barcelona, 2006. [60] W. Woess, Cogrowth of groups and simple random walks, Arch. Math. 41 (1983), pp. 363–370. [61] A. Zuk, On property (T) for discrete groups. Rigidity in dynamics and geometry (Cambridge, 2000), Springer, Berlin, 2002, pp. 473–482. Elizaveta Frenkel Moscow, Russia e-mail: [email protected] Alexei G. Myasnikov Department of Mathematics and Statistics McGill University Montreal, Quebec, Canada e-mail: [email protected] Vladimir N. Remeslennikov Omsk Branch of Mathematical Institute SB RAS 13 Pevtsova Street Omsk 644099, Russia e-mail: [email protected]

Combinatorial and Geometric Group Theory Trends in Mathematics, 119–147 c 2010 Springer Basel AG

Twisted Conjugacy for Virtually Cyclic Groups and Crystallographic Groups Daciberg Gon¸calves and Peter Wong Abstract. A group is said to have the property R∞ if every automorphism has an inﬁnite number of twisted conjugacy classes. In this paper, we classify all virtually cyclic groups with the R∞ property. Furthermore, we determine which of the 17 crystallographic groups of rank 2 have this property. Mathematics Subject Classiﬁcation (2000). Primary: 20E45; Secondary: 55M20. Keywords. Reidemeister number, elementary groups, Gromov hyperbolic groups, ﬁxed point theory.

1. Preliminaries Let ϕ : G → G be a group endomorphism and let R(ϕ) denote the Reidemeister number of ϕ, or equivalently the cardinality of the set of ϕ-twisted conjugacy classes, i.e., the number of orbits of the (left) action σ · α → σαϕ(σ)−1 where σ, α ∈ G. Our primary interest is in the (in)ﬁniteness of R(ϕ) for automorphisms ϕ. We say that G has the property R∞ for automorphisms, in short G has the property R∞ , if for every automorphism ϕ : G → G we have R(ϕ) = ∞. By elementary groups, we mean groups that are ﬁnite extensions of cyclic groups, i.e., these are the virtually cyclic groups. This work is motivated by Levitt and Lustig [7] who showed implicitly that automorphisms of ﬁnitely generated non-elementary Gromov hyperbolic groups have inﬁnite Reidemeister numbers (see also [3]). These results leave out the class of elementary Gromov hyperbolic groups. It is the purpose of this paper to study the R∞ property for the class of (inﬁnite) elementary groups or equivalently the This work was initiated during the ﬁrst author’s visit to Bates College April 8–30, 2006. Research was continued during the ﬁrst author’s visit to Bates College in the period March 12–16, 2008 and the second author’s subsequent visits to IME-USP in 2006, 2007, and 2008. The visits of the second author were supported by a grant from the National Science Foundation OISE-0334814. The second author was supported in part by the National Science Foundation DMS-0805968.

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class of virtually Z groups and their generalizations, namely, the family of virtually free abelian groups. The study of the inﬁniteness of R(ϕ) has application in ﬁxed point theory. For instance, examples of ﬁnitely generated torsion-free nilpotent groups with property R∞ have been constructed in [5] so that for every positive integer n ≥ 5, there exists a compact n-dimensional nilmanifold M n such that every homeomorphism of M n is isotopic to a ﬁxed point free map. Our basic technique in computing the Reidemeister number R(ϕ) in this paper is by means of short exact sequences as stated in the following lemma whose proof can be found in [4], for instance. Consider the following commutative diagram of short exact sequences of groups and endomorphisms. p

i

1 −−−−→ A −−−−→ B −−−−→ C −−−−→ 1 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ϕ$ ϕ$ ϕ $

(1.0.1)

p

i

1 −−−−→ A −−−−→ B −−−−→ C −−−−→ 1 There is a short exact sequence of sets and corresponding functions ˆi and pˆ: ˆi

pˆ

R(ϕ ) → R(ϕ) → R(ϕ)

(1.0.2)

where R(ψ) denotes the set of ψ-twisted conjugacy classes. Denote by R(ψ) the cardinality of R(ψ). Lemma 1.1. Given the commutative diagram labeled (1.0.1) above, (1) if R(ϕ) = ∞ then R(ϕ) = ∞, #ˆi(R(τα ϕ )) where τα (β) = αβα−1 and p(α) = α, ¯ (2) R(ϕ) = [α]∈R(ϕ) ¯ (3) if either F ix(τα ϕ ) = 1 for every [¯ α] ∈ R(ϕ), ¯ in which case ˆi is injective, or C is ﬁnite, and if R(α · ϕ ) = R(τα ϕ ) = ∞ for some [¯ α] ∈ R(ϕ) then we have R(ϕ) = ∞, (4) if C = Z, ϕ(t) = t−1 and either R(ϕ ) = ∞ or R(t · ϕ ) = ∞ then R(ϕ) = ∞. This paper is organized as follows. In Section 2, we classify the inﬁnite virtually cyclic groups with the R∞ property. Every inﬁnite virtually cyclic group G admits a short exact sequence 0 → Z → G → Q → 1 for some ﬁnite group Q. We show (cf. Prop. 2.8) that if this extension is not central then G has the R∞ property. On the other hand, if the extension is central, then we give necessary and suﬃcient conditions (cf. Prop. 2.5) for G to have R∞ property. In Section 3, we ﬁrst recall the 17 two-dimensional crystallographic groups in the order as listed in [1]. In Sections 4–7, we study the R∞ property for these groups. As it turns out, of the 17 cases, only cases 1, 2 (Section 3) and 13 (Section 6) do not possess the R∞ property. Concluding remarks are made in Section 8.

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2. Elementary groups In this section we study the R∞ property for the elementary groups. We will consider an alternative description of such groups and then we show that there are examples of such groups with the R∞ property. This is somewhat surprising since such groups are virtually Z and Z does not have the property R∞ for R(−id) = 2 where −id : Z → Z is the non-trivial automorphism. Such groups G have torsion, except the case when it is isomorphic to Z. So except in this case we do not have a manifold which has the homotopy type of a ﬁnite K(G, 1). Recall that the elementary groups are the ones which are virtually cyclic. If G is an inﬁnite then we have a short exact sequence 0→Z→G→Q→1

(2.0.3)

where Q is a ﬁnite group. Such groups have been completely classiﬁed (see, e.g., [2, Theorem 6.12, p. 129]. Next, we show that this classiﬁcation falls into two classes according to whether the extension (2.0.3) is central or not. Theorem 2.1. Let G be an inﬁnite virtually cyclic group which admits the extension (2.0.3). Then (i) the extension (2.0.3) is central iﬀ G ∼ = F Z for some ﬁnite normal subgroup F; (ii) the extension (2.0.3) is not central iﬀ there exists a ﬁnite normal subgroup F such that G/F ∼ = D∞ ∼ = Z2 ∗ Z2 . Proof. It follows from [2, Theorem 6.12, p. 129] that for every inﬁnite virtually cyclic group G, there exists a ﬁnite normal subgroup F such that G/F is either Z or the inﬁnite dihedral group D∞ . It suﬃces to show that G/F ∼ = Z iﬀ (2.0.3) is central. Suppose G/F ∼ = Z. Then G ∼ = F θ Z. Since F is ﬁnite, the automorphism θ(t) has ﬁnite order, say r, where Z ∼ = t. Now the inﬁnite cyclic subgroup H =

(1F , tr ) is central in G since (1F , tr )(α, tj ) = ((θ(t))r (α), tr+j ) = (α, tr+j ) = (α, tj )(1F , tr ). Moreover, G/H is ﬁnite and hence G admits a central extension of the form (2.0.3). Conversely, suppose G admits a central extension of the form (2.0.3). We write Q = {1Z, g1 Z, . . . , gk Z} with 1 = g0 . For any g ∈ G, there exists a unique i, 0 ≤ i ≤ k such that g = gi n where n ∈ Z. This gives rise to a map π : G → Z which is a well-deﬁned group homomorphism since Z is a central subgroup in G. Evidently, π is surjective and F = Ker π is ﬁnite. Since Z is free, π has a section and thus G ∼ = F Z. We will divide the family of inﬁnite elementary groups into two families. Let F1 be the family of the groups such that the extension (2.0.3) is central and F2 otherwise. Corollary 2.2. Let G be an inﬁnite elementary group. Then either G = Z or G has torsion.

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Proof. Follows immediately from Theorem 2.1.

Now we move to the case where the extension is not central, i.e., G ∈ F2 . The following simple lemma gives an alternative description of such groups. Lemma 2.3. Let G ∈ F2 . Then G is the middle term of a short exact sequence of the form p

1 → G1 → G → Z2 → 0 where G1 ∈ F1 . Furthermore, if w ∈ G such that p(w) is the nontrivial element of Z2 then w2 belongs to the torsion subgroup of G1 . Proof. Given an extension 0 → Z → G → Q → 1, consider Q1 ⊂ Q the subgroup consisting of elements which act trivially on Z. Then the pullback of this inclusion is a subgroup G1 of G of index 2 so that the sequence 1 → G1 → G → Z2 → 0 is exact. Moreover, the subgroup G1 yields a short exact sequence of the form 0 → Z → G1 → Q1 → 1 and thus G1 ∈ F1 . For the second part, consider the short exact sequence given from the ﬁrst part. By Theorem 2.1(i), G1 = H φ Z. Now w2 is of the form (h, t ). We will show that = 0. Suppose that wk ∈ Z for some k = 0. Since w commutes with wk it follows that w acts trivially on the subgroup (k)Z, which is a contradiction if = 0 because w acts as multiplication non-trivially on some inﬁnite subgroup of Z. So it remains to show that wk ∈ Z for some k = 0. Let r be the order of the automorphism φ(t) ∈ Aut(H) where Z = t. A simple calculation shows that w2r = (h · φ(t) (h) · · · · · φ(t)(r−1) (h), tr ). Note that the element h · φ(t) (h) · · · · · φ(t)(r−1) (h) ∈ H has ﬁnite order, say s. Since φ(t)r = id it follows that w2rs = (h · φ(t) (h) · · · · · φ(t)(r−1) (h))s , trs ) = (1, trs ) so that wk ∈ Z for some k = 0. Now the proof is complete. The above lemma provides the following result: Proposition 2.4. If G ∈ F2 then there exist a ﬁnite subgroup H ⊂ G, an embedding Z ⊂ G, such that H ∩Z = 1, and H and Z generate G. Further, there is a subgroup H1 ⊂ H of index 2 and a nontrivial subgroup H ⊂ Z which is normal and the action G → Aut(Z) is nontrivial. Proof. Straightforward.

Now we give a necessary and suﬃcient condition for a group in F1 to have the R∞ property. Proposition 2.5. Let Z = t, H a ﬁnite group, θ(t) ∈ Aut(H), and G = H θ Z. Then G has the R∞ property if, and only if, θ(t) is not conjugated to its inverse in Aut(H).

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Proof. Assume that G has the R∞ property. Suppose that there is an automorphism φ ∈ Aut(H) such that φ ◦ θ(t) ◦ φ−1 = θ(t−1 ). Let Φ(h, tr ) = (φ(h), t−r ) for h ∈ H and tr ∈ Z. Note that Φ((1, t)(h, 1)) = Φ(θ(t)(h), t) = (φ(θ(t)(h)), t−1 ) = (θ(t−1 )(φ(h)), t−1 ) −1

= (1, t

(2.0.4)

)(θ(h), 1) = Φ(1, t)Φ(h, 1).

It follows that Φ ∈ Aut(G) such that φ = Φ|H and the induced automorphism Φ is given by t → t−1 . Since H is ﬁnite and R(Φ) = 2, it follows from Lemma 1.1(2) that R(Φ) < ∞, a contradiction. Hence, such an automorphism φ cannot exist. Conversely, we assume that there is no automorphism φ ∈ Aut(H) with φ ◦ θ(t) ◦ φ−1 = θ(t−1 ). Let Φ : G → G be an automorphism. Since H is the unique maximal torsion subgroup, it is characteristic in G. Let ϕ = Φ|H . The induced homomorphism Φ on G/H = Z is either the identity or given by t → t−1 . If Φ(t) = t−1 then the calculation in (2.0.4) shows that for all h ∈ H, ϕ(θ(t)(h)) = θ(t−1 )(ϕ(h)) since Φ is a group homomorphism. This violates the assumption on Aut(H) and thus Φ must be the identity on G/H = Z and hence R(Φ) = ∞. It follows from Lemma 1.1(1) that G has the R∞ property. As before let G ∈ F1 . Corollary 2.6. If Aut(H) is abelian and θ is neither trivial nor of order 2, then G has the R∞ property. Proof. If Aut(H) is abelian the equality φ ◦ θ(t) ◦ φ−1 = θ(t−1 ) implies θ is either trivial or of order 2. Hence the result follows. Example 2.7. Consider the group G = Z5 Z where the action Z → Aut(Z5 ) = Z4 is the natural projection. This group is also an extension of Z by a ﬁnite group (it is an elementary hyperbolic group), namely 0 → 4Z → Z5 Z → Z5 Z4 → 0 where the ﬁrst map is the inclusion in the second component. By the corollary above the group G has the R∞ property. Now we will consider groups in F2 . We ﬁrst show an example of a group in F2 which has the property R∞ . This group is Z Z2 which is also isomorphic to Z2 ∗ Z2 and this group has been shown to have the R∞ property [5, Theorem 2]. Because this group is the fundamental group of the closed 3-manifold RP 3 #RP 3 , it follows that for every homotopy equivalence f of RP 3 #RP 3 the Reidemeister number of f is always inﬁnite. Now we derive the main result about the groups in F2 . Proposition 2.8. Every group G ∈ F2 has the R∞ property.

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Proof. Let p : G → Q be the projection given by a short exact sequence which deﬁnes G and Q1 ⊂ Q the subgroup of index 2 of the elements which acts trivially on Z. By Lemma 2.3, we have the following short exact sequence 0 → G1 = p−1 (Q1 ) → G → Z2 → 0, where G1 ∈ F1 is of the form H Z for some ﬁnite H. Let us assume that G1 is characteristic. If so, given any automorphism φ : G → G we have that the restriction of φ to G1 is also an automorphism and so φ(H) = H. Therefore φ : G → G induces an automorphism on the quotient φ¯ : G/H → G/H where G/H = Z Z2 . Since Z Z2 has the property R∞ , it follows that G also has the property and the result follows. It remains to show that G1 is characteristic. For this, let w ∈ G be an element which projects to the non-trivial element in Z2 . It suﬃces to show that φ(w) also projects to the non-trivial element. Suppose not, i.e., φ(w) = v ∈ G1 , and let k ∈ Z(G1 ) ∩ Z(φ(G1 )) be a nontrivial element of inﬁnite order (which is possible because both groups have ﬁnite index in G). Then applying the automorphism φ to the equation t ◦ k ◦ t−1 = −k leads to a contradiction.

3. Crystallographic groups The crystallographic groups of rank 2, or the so-called wallpaper groups, are the middle term of extensions of Z2 by a ﬁnite group. So they are special cases of virtually free abelian groups of rank 2. The main references that we are going to use for these groups are Coexter and Moser [1] pages 40 to 51 and [8]. For each of the crystallographic groups we will provide the presentation given by [1] and in most cases also the one given by [8]. The label of the item corresponds to the enumeration given in [1]. For our computation, it is sometimes more convenient to use alternative presentations some of which are given in [8]. The 17 wallpaper groups are listed as follows. 1. X, Y | XY = Y X. This group corresponds to the group G1 and the presentation given in [8] is the same. 2. X, Y, T | XY = Y X, T 2 = 1, T XT = X −1 , T Y T = Y −1 . This group corresponds to the group G2 and the presentation given in [8] is the same. 3. X, Y, R| XY = Y X, R2 = 1, RXR = X −1, RY R = Y . This group corresponds to the group G11 and the presentation given in [8] is the same. 4. X, Y, P | XY = Y X, P 2 = Y, P −1 XP = X −1 . This group corresponds to the group G31 and the presentation given in [8] is the same. 5. P, Q, R| P 2 = Q2 , R2 = 1, RP R = Q. This group corresponds to the group G21 . We need several steps to show that the two presentations are equivalents. 6. X1 , X2 , X3 , Y |X12 = X22 = X32 = 1, X1 Y = Y X1 , X2 Y = Y X2 , X1 X3 = X3 X1 , X2 X3 = X3 X2 , X3 Y X3−1 = Y −1 . According to [1], this group is isomorphic to D∞ × D∞ where D∞ is the inﬁnite dihedral group.

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7. P, Q, R| P 2 = Q2 , R2 = 1, RP R = P −1 , RQR = Q−1 . By letting P = αβ and Q = β, this group can be given the following presentation:

α, β, R|βαβ −1 = α−1 , αR = α, β R = β −1 , R2 = 1. 8. P, Q, T | P 2 = Q2 , T 2 = 1, T P T = Q−1 . By letting P = αβ and Q = β, this group can be given the following presentation:

α, β, T |βαβ −1 = α−1 , αT = α−1 , β T = αβ −1 , T 2 = 1. 9. R1 , R2 , T | R12 = R22 = T 2 = (R1 R2 )2 = (R1 T R2 T )2 = 1. By letting α = R2 T R1 and β = R1 R2 T , this group can be given the following presentation and is a ﬁnite extension of Z2 : ! α, β, R1 , R2 |αβ = βα,αR1 = β, β R1 = α, αR2 = β −1 , " β R2 = α−1 , R12 = R22 = (R1 R2 )2 = 1 . 10. S, T | S 4 = T 2 = (ST )4 = 1. This group corresponds to the group G4 . To see that the two presentations are equivalent, consider the map σ → S, α → T S 2 and β → ST S. It is a straightforward calculation to see that this bijection between the sets of generators extends to an automorphism of the group. 11. R, R1 , R2 | R2 = R12 = R22 = (RR1 )4 = (R1 R2 )2 = (R2 R)4 = 1. This group corresponds to the group G14 and the isomorphism is given by σ → RR1 , α → R2 RR1 R, β → RR2 RR1 , and ρ → R1 . 12. R, S| R2 = S 4 = (RS −1 RS)2 = 1. This group corresponds to the group G24 and the isomorphism is given by σ → S, ρ → RS −1 , α → RS −1 RS −1 , and β → SRS −1 RS −2 . 13. S1 , S2 , S3 | S13 = S23 = S33 = S1 S2 S3 = 1. This group corresponds to the group G3 . To see that the two presentations are equivalent, consider the map S1 → σ, S2 → ασα and S3 → βσβ. It is a straightforward calculation to see that this bijection between the sets of generators extends to an automorphism of the group. 14. R, S| R2 = S 3 = (RS −1 RS)3 = 1. This group corresponds to the group G13 and the isomorphism is given by R → ρ and S → σαβ −1 . 15. R1 , R2 , R3 | R12 = R22 = R32 = (R1 R2 )3 = (R2 R3 )3 = (R3 R1 )3 = 1. This group corresponds to the group G23 . The isomorphism between them is given by R1 → ρσβ, R2 → σρ, R3 → ρ. 16. S, T | S 3 = T 2 = (ST )6 = 1. This group corresponds to the group G6 . To see that the two presentations are equivalent, consider the map σ → ST , α → T ST ST ST and β → S 2 T ST . It is a straightforward calculation to see that this bijection between the sets of generators extends to an automorphism of the group. One can also construct the inverse map S → σ 4 α−1 , T → ασ3 . 17. R, R1 , R2 | R2 = R12 = R22 = (R1 R2 )3 = (R2 R)2 = (RR1 )6 = 1. This group corresponds to the group G16 . The correspondence is given by σ → R1 R, α → R2 R(R1 R)3 , β → R1 R2 (R1 R)2 , and ρ → R.

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4. Cases 1–5 For Cases 1–5, these are the wallpaper groups that can be expressed as extensions of Z2 by Z2 or by the trivial group. 4.1. Case 1. It is well known (see, e.g., [6]) that for endomorphisms ψ : Zn → Zn , R(ψ) < ∞ iﬀ det(1 − ψ) = 0 and in the case R(ψ) < ∞, R(ψ) = | det(1 − ψ)|. The group Z2 does not have property R∞ since one can easily ﬁnd automorphism ϕ such that det(1 − ϕ) = 0. This implies that R(ϕ) < ∞. 4.2. Case 4. This group K is the fundamental group of the Klein bottle which is doubly-covered by the torus. It has been shown in [5] that K has the property R∞ . For cases 2, 3, and 5, each can be expressed as a semi-direct Z2 θ Z2 . Since every element in a semi-direct product can be uniquely written as a product of elements from the kernel and from the quotient, every automorphism ϕ can be represented by an array of the form ⎤ ⎡ a c r (4.2.1) ϕ = ⎣b d s⎦ δ γ where ϕ(α) = αa β b t ; ϕ(β) = αc β d tδ ; ϕ(t) = αr β s tγ . Here, Z2 ∼ = α, β|αβ = βα and Z2 = t|t2 = 1. 4.3. Case 2. This group corresponds to G2 in [8] and has the following presentation G = α, β, t|αβ = βα, αt = α−1 , β t = β −1 , t2 = 1. Note that G ∼ = Z2 θ Z2 where θ : Z2 → Aut(Z2 ) is given by

−1 0 . θ(t) = 0 −1 Let ϕ ∈ Aut(G) be given by

⎤ 1 2 0 ϕ = ⎣−1 −1 0⎦ . 0 0 1 ⎡

It is easy to see that ϕ is an automorphism and it induces the identity ϕ¯ = id on Z2 with two Reidemeister classes [1] and [t]. Over the class [1], R(ϕ ) = | det(1−ϕ )| = * ) 1 2 2 where ϕ = . Over the class [t], R(t · ϕ ) = | det(1 − θ(t)ϕ )| = 2. −1 −1 It follows from Lemma 1.1(3) that R(ϕ) = 4 < ∞. Hence, G does not have property R∞ .

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4.4. Case 3. This group corresponds to G11 in [8] and has the following presentation G = α, β, t|αβ = βα, αt = α, β t = β −1 , t2 = 1. Thus, G ∼ = Z2 θ Z2 where θ : Z2 → Aut(Z2 ) is given by

1 0 . θ(t) = 0 −1 Let ϕ ∈ Aut(G) be given by (4.2.1). Since t is of order 2, it follows that γ = 1. Moreover, ϕ(t2 ) = 1 implies that (αr β s t)2 = αr β s αr β −s = α2r = 1. Thus, r = 0. Since ϕ(β t ) = ϕ(β −1 ), we have (β s t)(αc β d tδ )(β s t) = t−δ β −d α−c .

(4.4.1)

Since α is central in G, by equating the exponents of α on both sides of (4.4.1) yields that c = 0. Now c = 0 and r = 0, the subgroup H = β, t|β t = β −1 , t2 = 1 ∼ = Z Z2 ∼ = D∞ is characteristic in G. By [5], H = D∞ has the property R∞ . It follows from Lemma 1.1(3) that G also has property R∞ . 4.5. Case 5. This group corresponds to G21 in [8] and has the following presentation G = α, β, t|αβ = βα, αt = β, β t = α, t2 = 1. Thus, G ∼ = Z2 θ Z2 where θ : Z2 → Aut(Z2 ) is given by

0 1 . θ(t) = 1 0 Let ϕ ∈ Aut(G). Again, since t is of order 2, we have γ = 1. Since ϕ(t2 ) = 1, it follows that (αr β s t)2 = αr β s β r αs = (αβ)r+s = 1. Thus, r + s = 0. The equality ϕ(αt ) = ϕ(β) yields αr β s tαa β b t αr β s t = αc β d tδ . (4.5.1) By equating the exponents of t, we have = δ. If = δ = 1, then ϕ(αβ) = ϕ(βα) yields αa β b tαc β d t = αc β d tαa β b t ⇒ αa β b β c αd = αc β d β a αb ⇒ a + d = b + c. r s a

In fact, (4.5.1) yields α β β αb αr β s = αc β d so that b+2r = c and a+2s = d. Since r + s = 0, we have a + b = c + d. It follows that a = c and b = d, a contradiction to the assumption that ϕ is an automorphism. The case = δ = 1 cannot occur. Suppose = δ = 0. In this case, (4.5.1) becomes αr β s tαa β b αr β s t = αc β d or αr β s β a αb β r αs = αc β d . It follows that r + b + s = c and s + a + r = d so that b = c and a = d. Thus, the subgroup generated by α and β, which is isomorphic

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* ) a b . Since ϕ to Z , is characteristic and ϕ = ϕ|Z2 is given by the matrix b a is an automorphism, we have a2 − b2 = ±1. This happens when a = ±1, b = 0 or a = 0, b = ±1. In the latter case, | det(1 − ϕ )| = 0 so that R(ϕ ) = ∞ and hence R(ϕ) = ∞ by Lemma 1.1(3). In the former case, i.e., a = ±1, b = 0, | det(1 − t · ϕ )| = 0 so that R(t · ϕ ) = ∞ and hence R(ϕ) = ∞ by Lemma 1.1(3). Hence, G has property R∞ .

2

5. Cases 6–9 Next, we deal with cases 6–9. First, we study the cases 7 and 8 since in each case, the group is a semi-direct product of K, the fundamental group of the Klein bottle, and Z2 . 5.1. Case 7. This group can be given the following presentation G = α, β, t|βαβ −1 = α−1 , αt = α, β t = β −1 , t2 = 1 and G = K Z2 where K = α, β|βαβ −1 = α−1 and Z2 = t|t2 = 1. Let ϕ ∈ Aut(G). Since K is also a semi-direct product Z Z, it follows that ϕ can also be represented by ⎤ ⎡ a c r (5.1.1) ϕ = ⎣b d s⎦ . δ γ Note that γ = 1 since ϕ(t) has order 2. Moreover, (ϕ(t))2 = 1 implies that s αr β s tαr β s t = 1 or αr β s αr β −s = αr α(−1) r = 1. If s is even then r = 0. Suppose (, δ) = (0, 1). The equality ϕ(αt ) = ϕ(α) yields αr β s αa β −b β −s α−r = αa β b ⇒ αr (β s αa β −s )β −b α−r = αr α(−1) a β −b α−r = αa β b s

⇒ αr+(−1) a β −b = αa β b αr = α(−1) s

b

r+a b

β .

It follows that b = 0 and either a = 0 or s is even. If a = 0, then we have ϕ(α) = 1, a contradiction. This means that s is even and a = 0. In this case, the equality ϕ(βαβ −1 ) = −1 ϕ(α ) yields αc β d tαa β b t−1 β −d α−c = β −b α−a ⇒ αc β d αa β −b β −d α−c = β −b α−a ⇒ αc β d αa β −d β −b α−c = β −b α−a ⇒ αc α(−1) a α−c = α−a d

(here b = 0).

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It follows that d is odd since a = 0. Now ϕ(β t ) = ϕ(β −1 ) implies that αr β s tαc β d tt−1 β −s α−r = t−1 β −d α−c ⇒ αr β s αc β −d β s α−r = β d α−c ⇒ αr αc β 2s−d α−r = β d α−c d

⇒ αr+c β 2s−d = α(−1)

(here s is even)

(r−c) d

β = αc−r β d

(here d is odd).

It follows that s = d, a contradiction to the fact that s is even and d is odd. Therefore, the case (, δ) = (0, 1) cannot occur. Suppose that (, δ) = (1, 0). The equality ϕ(βαβ −1 ) = ϕ(α−1 ) yields αc β d αa β b tβ −d α−c = t−1 β −b α−a ⇒ αc β d αa β b β d α−c = β b α−a ⇒ αc β d αa β d β b = β b αc−a ⇒ αc β d αa β d = β b αc−a β −b d

b

⇒ αc α(−1) a β 2d = α(−1)

(c−a)

.

b

This implies that d = 0 and c + (−1)d a = (−1) (c − a). If b is even then a = 0 and if b is odd then c = 0, which in this case implies that ϕ(β) = 1, a contradiction. Thus, d = 0, b is even, and a = 0. Now ϕ(β t ) = ϕ(β −1 ) implies that αr β s tαc t−1 β −s α−r = α−c ⇒ αr β s αc β −s α−r = α−c s

⇒ αr α(−1) c α−r = α−c . Since c = 0, it follows that s must be odd. Now, ϕ(αt ) = ϕ(α) yields αr β s tβ b β −s α−r = β b t ⇒ αr β s β −b β s α−r = β b ⇒ αr β 2s−b α−r = β b . It follows that s = b. But b is even and s is odd, a contradiction. Thus, the case (, δ) = (1, 0) cannot occur. Suppose (, δ) = (1, 1). The equality ϕ(αt ) = ϕ(α) yields αr β s tαa β b β −s α−r = αa β b t ⇒ αr β s αa β −b β s α−r = αa β b s

b

⇒ αr+(−1) a β 2s−b = αa β b αr = αa+(−1) r β b . s

It follows that s = b and r + (−1) a = (−1)b r + a. If s is odd then r = a. This case cannot occur otherwise ϕ(α) = ϕ(t), a contradiction. Thus, s = b and s is even.

130

D. Gon¸calves and P. Wong Now, ϕ(β t ) = ϕ(β −1 ) implies that αr β s tαc β d tt−1 β −s α−r = t−1 β −d α−c ⇒ αr β s αc β −d β s α−r = β d α−c ⇒ αr β s αc β −s β 2s−d = β d αr−c s

⇒ αr α(−1) c β 2s−d = β d αr−c d

⇒ αr+c β 2s−2d = α(−1)

(r−c)

.

It follows that s = d and c = 0. Thus, in this case, we have b = d = s is even and c = 0. The equality ϕ(βαβ −1 ) = ϕ(α−1 ) yields β d tαa β b tt−1 β −d = t−1 β −b α−a ⇒ β d αa β −b β d α−c = β b αa β −b β b α−c = β b α−a b

⇒ α(−1) a β b = α(−1)

b

(−a) b

β .

It follows that a = 0. This means that ϕ(α) = ϕ(β), a contradiction. Hence, (, δ) = (0, 0) and K is characteristic. Since K has property R∞ , it follows that G also has property R∞ . 5.2. Case 8. This group can be given the following presentation G = α, β, t|βαβ −1 = α−1 , αt = α−1 , β t = αβ −1 , t2 = 1 and G = K Z2 where K = α, β|βαβ −1 = α−1 and Z2 = t|t2 = 1. Let ϕ ∈ Aut(G). As in Case 7, the automorphism ϕ can be represented by (5.1.1). Since t2 = 1, we may assume that γ = 1. Moreover, ϕ(t2 ) = 1 implies that (αr β s t)2 = 1 so that αr β s α−r β −s β s (αβ −1 )s = 1. When s is odd, we have (αβ −1 )s = β 1−s αβ −1 . It follows that αr β s α−r β −s = α and so 2r = 1, a contradiction. Thus, s must be even, in which case, (αβ −1 )s = β −s . The equality ϕ(β t ) = ϕ(αβ −1 ) yields αr β s tαc β d tδ t−1 β −s α−r = αa β b t t−δ β −d α−c .

(5.2.1)

By equating the exponents of t, we have δ = − δ which implies that = 0. If δ = 0, then K is characteristic and G will have property R∞ following the same argument as in Case 7. Thus, we assume δ = 1. The equality ϕ(αt ) = ϕ(α−1 ) yields αr β s tαa β b t−1 β −s α−r = β −b α−a which implies that αr β s α−a (αβ −1 )b β −s α−r = β −b α−a . If b is odd then (αβ −1 )b = αβ −b so that αr β s α−a αβ −b β −s α−r = β −b α−a . Since b is odd and s is even, it follows that αr−a+1 β −b αa−r = β −b such that 2a = 2r + 1, a contradiction. Thus, b must be even.

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131

Now, the equality ϕ(βαβ −1 ) = ϕ(α−1 ) yields αc β d tαa β b t−1 β −d α−c = β −b α−a ⇒ αc β d α−a (αβ −1 )b β −d α−c = β −b α−a ⇒ αc β d α−a β −b−d α−c = β −b α−a c d −a −d

⇒α β α

β

d −a −d

⇒β α

β

(since b is even)

c−a

=α

= α−a .

If a = 0 then d must be even. Suppose d is even. Then ϕ(β 2 ) = (αc β d t)2 = αc β d α−c β −d = αc α−c = 1, a contradiction to the assumption that ϕ is an automorphism and β is of inﬁnite order. Thus, d must be odd and hence a = 0. Now that a = 0, (5.2.1) becomes αr β s tαc β d−s α−r = β b t−1 β −d α−c ⇒ αr β s α−c (αβ −1 )d−s αr t = β b (αβ −1 )−d αc t ⇒ αr β s α−c (αβ −1 )d−s−1 (αβ −1 )αr = β b (αβ −1 )−d−1 (αβ −1 )αc ⇒ αr β s α−c β 1+s−d αβ −1 αr = β b+d βαβ −1 αc (both d − s − 1 and −d − 1 are even) ⇒ αr α−c β 2s−d βαβ −1 αr = β b+d αc−1 ⇒ αr−c β 2s−b−2d = αr−c

(since b is even and d is odd).

Thus, 2s = b + 2d. Since d is odd, we have ϕ(β 2 ) = (αc β d t)2 = αc β d α−c (αβ −1 )d which in turn is equal to α2c−1 . Now, we have ⎤ ⎡ 0 c r ϕ = ⎣2(s − d) d s⎦ . 0 1 1 Since ϕ(β 2 ) = α2c−1 , the subgroup by α and β 2 is ϕ-invariant and * ) H generated 0 2c − 1 where κ = s − d. Note that H is H ∼ = Z2 . If ϕ = ϕ|H then ϕ = κ 0 a subgroup of index 2 in K so H is a subgroup of ﬁnite index in G. Since H is generated by α and β 2 , it is easy to see that αη and (β 2 )η belong to H for η = α, β, t. This means that H is a normal subgroup of ﬁnite index in G so that there is a short exact sequence 0→H →G→F →1 for some ﬁnite group F . Since ϕ is an automorphism det ϕ = ±1 which means that (2c − 1)κ = ±1, we have (c, κ) ∈ {(0, 1), (0, −1), (1, 1), (1, −1)}. When (c, κ) = (0, −1) or (c, κ) = (1, 1), it is easy to see that det(1−ϕ ) = 0 so that R(ϕ ) = ∞. Note that β ∈ G−H ¯ we so the projection β¯ of β in F is non-trivial. Over the Reidemeister class [β],

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consider R(τ * β ϕ ) where τβ ϕ is simply ϕ conjugated by β. It follows that τβ ϕ = ) 0 1 − 2c . Now, for (c, κ) = (0, 1) or (c, κ) = (1 − 1), det(1 − τβ ϕ ) = 0 so that κ 0 R(τβ ϕ ) = ∞ for either case and hence R(ϕ) = ∞ by Lemma 1.1(3). Thus when δ = 1, R(ϕ) = ∞. We can now conclude that G has property R∞ .

5.3. Case 6. 1 2 According to [1], this group is isomorphic to D∞ ×D∞ . Write G = D∞ ×D∞ where i ∼ 1 s −1 2 2 t −1 2 D∞ = D∞ , D∞ = α, s|α = α , s = 1, and D∞ = β, t|β = β , t = 1. 1 2 Consider the subgroup H = α, β|αβ = βα ⊂ D∞ × D∞ . Let ϕ ∈ Aut(G). 1 i Every element in D∞ can be uniquely written as α s , ∈ {0, 1} and likewise every 2 element in D∞ is of the form β j tδ , δ ∈ {0, 1}. Suppose ϕ(α, 1) = (αa s , β b tδ ) ˜ and ϕ(1, β) = (αc s˜, β d tδ ). We may assume that + δ = 0, ˜ + δ˜ = 0, δ = 0, ˜ = (1, 0) otherwise ϕ(α, β) = (αA s, β B t) for ˜δ˜ = 0. If (, δ) = (1, 0) then (˜ , δ) some A, B. However, αA s and β B t are both of ﬁnite order which contradicts that H is torsion-free. In this case, ϕ(α2 , 1) = (1, β 2b ) and ϕ(1, β 2 ) = (1, β 2d ) so the subgroup generated by ϕ(α2 , 1) and ϕ(1, β 2 ) is of rank 1, a contradiction to the fact that ϕ is an automorphism and (α2 , 1) and (1, β 2 ) together generate Z2 . ˜ cannot occur. Note that the case Similarly, the case where (, δ) = (0, 1) = (˜ , δ) A ˜ (, δ) = (1, 1) = (˜ , δ) cannot occur since α s and β B t are both of ﬁnite order for ˜ and thus H is characteristic. any A and B. This implies that (, δ) = (0, 0) = (˜ , δ) ∼ Now, G = H θ (Z2 × Z2 ) where Z2 × Z2 = s|s2 = 1 × t|t2 = 1

1 0 −1 0 . θ(1, t) = θ(s, 1) = 0 −1 0 1 Let ϕ be the induced automorphism on the quotient Z2 × Z2 . Suppose ϕ(s, 1) = (1, t) and ϕ(1, t) = (s, 1). The equality ϕ(αs ) = ϕ(α−1 ) yields and

(1, t)(αa , β b )(1, t) = (α−a , β −b ) ⇒ a = 0. Similarly, the equality ϕ(β t ) = ϕ(β −1 ) yields d = 0. This implies that ϕ = * ) 0 c . It follows that (b, c) ∈ {(1, 1), (−1, −1), (1, −1), (−1, 1)}. If (b, c) = (1, 1) b 0 or (−1, * −1), det(1−ϕ ) = 0 ⇒ R(ϕ ) = ∞. If (b, c) = (1, −1) or (−1, 1), θ(s, 1)ϕ = ) 0 −c and so det(1 − θ(s, 1)ϕ ) = 0 ⇒ R(θ(s, 1)ϕ ) = ∞. Either case implies b 0 that R(ϕ) = ∞ by Lemma 1.1(3). Suppose ϕ(s, 1) = (s, 1) and ϕ(1, t) = (1, t). The equality ϕ(β t ) = ϕ(β −1 ) yields (1, t)(αc , β d )(1, t) = (α−c , β −d ) ⇒ c = 0. Similarly, using the equality ϕ(αs ) = ϕ(α−1 ), we can show that b = 0 so that * ) a 0 . ϕ = 0 d

Twisted Conjugacy Classes

133

In this case, ad = det ϕ = ±1 so that (a, d) ∈ {(1, 1), (−1, −1), (1, −1), (−1, 1)}. Similar calculations as above show that R(ϕ) = ∞. Suppose ϕ(s, 1) = (s, t). The equality ϕ(β s ) = ϕ(β) implies that (s, t)(αc , β d )(s, t) = (αc , β d ) ⇒ c = 0 = d. This leads to a contradiction. Likewise, if ϕ(1, t) = (s, t) then the equality ϕ(αt ) = ϕ(α) implies that (s, t)(αa , β b )(s, t) = (αa , β b ) ⇒ a = 0 = b,

a contradiction.

Hence, we conclude that R(ϕ) = ∞ and G has the property R∞ . 5.4. Case 9. This group has the following presentation: G = R1 , R2 , T | R12 = R22 = T 2 = (R1 R2 )2 = (R1 T R2 T )2 = 1. Consider the subgroup H generated by R1 , R2 , R1 T R1 T, R2 T R2 T . If we let σ = R1 , ρ = R2 , α = R1 T R1 T and β = R2 T R2 T , then H = α, β, σ, ρ|αβ = βα, ασ = α−1 , β σ = β, αρ = α, β ρ = β −1 , (σρ)2 = 1 = σ 2 = ρ2 . Note that H ∼ = D∞ × D∞ and G = H θ T |T 2 = 1 where θ(T ) is given by −1 α → α ; β → β −1 ; σ → α−1 σ; and ρ → β −1 ρ. Since G = H θ Z2 , we have G = α, β, σ, ρ, T |αβ = βα, ασ = α−1 , β σ = β, αρ = α, β ρ = β −1 , σ2 = ρ2 = (σρ)2 = T 2 = 1, αT = α−1 , β T = β −1 , σT = α−1 σ, ρT = β −1 ρ Let ϕ ∈ Aut(G). We can represent ϕ ⎡ a ⎢b ⎢ ϕ=⎢ ⎢ ⎣m f

by c d δ n g

r x s y γ z p q u v

⎤ A B⎥ ⎥ C⎥ ⎥, D⎦ E

where the columns correspond to the images under ϕ of the generators α, β, σ, ρ and T , respectively. The equality ϕ(σ T ) = ϕ(α−1 σ) yields αA β B σ C ρD T E αr β s σ γ ρp T u αA β B σ C ρD T E = T −f ρ−m σ − β −b α−a αr β s σ γ ρp T u . Equating the exponents of T , we have E + u + E = −f + u T

Similarly, ϕ(ρ ) = ϕ(β

−1

⇒

−f = 2E = 0

⇒

f = 0.

ρ) yields

α β σ ρ T α β σ ρ T v αA β B σ C ρD T E = T −g ρ−n σ −δ β −d α−c αx β y σ z ρq T v . A B C D

E

x y z q

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D. Gon¸calves and P. Wong

Equating the exponents of T , we have E + v + E = −g + v

⇒

−g = 2E = 0

⇒

g = 0.

2

Suppose u = 1. The equality ϕ(σ ) = 1 yields αr β s σ γ ρp T αr β s σ γ ρp T = 1 ⇒ αr β s σ γ ρp α−r β −s (α−1 σ)γ (β −1 ρ)p = 1.

(5.4.1)

Case (γ, p) = (0, 1): equation (5.4.1) becomes αr β s ρα−r β −s β −1 ρ = 1 which implies that αr β s α−r β s+1 = 1 or 2s + 1 = 0, a contradiction. Case (γ, p) = (1, 0): equation (5.4.1) becomes αr β s σα−r β −s α−1 σ = 1 which implies that αr β s αr β −s α = 1 or 2r + 1 = 0, a contradiction. Case (γ, p) = (1, 1): equation (5.4.1) becomes αr β s σρα−r β −s α−1 σβ −1 ρ = 1 which implies that αr β s σα−r β s α−1 σβ = 1 or αr β s αr β s αβ = 1 so that 2r + 1 = 0, a contradiction. Thus, when u = 1, we must have (γ, p) = (0, 0). With (γ, p, u) = (0, 0, 1), the equality ϕ(ασ ) = ϕ(α−1 ) yields αr β s T αa β b σ ρm αr β s T = ρ−m σ − β −b α−a ⇒ αr β s α−a β −b (α−1 σ) (β −1 ρ)m α−r β −s = ρ−m σ − β −b α−a .

(5.4.2)

Case (, m) = (0, 1): equation (5.4.2) becomes αr β s α−a β −b β −1 ρα−r β −s = ρβ −b α−a which implies that αr β s α−a β −b β −1 α−r β s = β b α−a . Thus, s − b − 1 + s = b. It follows that 2s = 2b + 1, a contradiction. Case (, m) = (1, 0): equation (5.4.2) becomes αr β s α−a β −b α−1 σα−r β −s = σβ −b α−a which implies that αr β s α−a β −b α−1 αr β −s = β −b αa . Thus, r − a − 1 + r = a or 2r = 2a + 1, a contradiction. Case (, m) = (1, 1): equation (5.4.2) becomes αr β s α−a β −b α−1 σβ −1 ρα−r β −s = ρσβ −b α−a which implies that αr β s α−a β −b α−1 σβ −1 α−r β s = σβ b α−a or αr β s α−a β −b α−1 β −1 αr β s = β b αa . Thus, r − a − 1 + r = a or 2r = 2a + 1, a contradiction. Thus, when u = 1, γ = p = = m = 0. Again, when u = 1, the equality ϕ(β σ ) = ϕ(β) yields αr β s T αc β d σ δ ρn T β −s α−r = αc β d σ δ ρn ⇒ αr β s α−c β −d (α−1 σ)δ (β −1 ρ)n β −s α−r = αc β d σ δ ρn .

(5.4.3)

Twisted Conjugacy Classes

135

Case (δ, n) = (0, 1): equation (5.4.3) becomes αr β s α−c β −d β −1 ρβ −s α−r = αc β d ρ which implies that αr β s α−c β −d β −1 β s α−r = αc β d . Thus, s − d − 1 + s = d or 2s = 2d + 1, a contradiction. Case (δ, n) = (1, 0): equation (5.4.3) becomes αr β s α−c β −d α−1 σβ −s α−r = αc β d σ which implies that αr−c−1 β s−d σβ −s α−r = αc β d σ or αr−c−1 β s−d β −s αr = αc β d . Thus, r − c − 1 + r = c or 2r = 2c + 1, a contradiction. Case (δ, n) = (1, 1): equation (5.4.3) becomes αr β s α−c β −d α−1 σβ −1 ρβ −s α−r = αc β d σρ which implies that αr−c−1 β s−d σβ −1 β s α−r = αc β d σ or αr−c−1 β s−d β s−1 αr = αc β d . Thus, r − c − 1 + r = c or 2r = 2c + 1, a contradiction. Thus, if u = 1 then γ = p = = m = δ = n = f = g = 0. Assuming u = 1, the equality ϕ(αρ ) = ϕ(α) yields αx β y σ z ρq T v αa β b αx β y σ z ρq T v = αa β b .

(5.4.4)

Case (q, v) = (0, 0): Note that z = 0 otherwise ϕ(ρ) would have inﬁnite order. Now z = 1, equation (5.4.4) becomes αx β y σαa+x β b+y σ = αa β b which implies that αx β y α−a−x β b+y = αa β b . It follows that a = 0 and y = 0. In this case, ϕ(β ρ ) = ϕ(β −1 ) yields αx σαc β d αx σ = β −d α−c ⇒ αx α−c β d αx = β −d α−c

⇒

d = 0.

⇒

c = 0.

Now ϕ(β σ ) = ϕ(β) yields αr β s T αc T β −s α−r = αc ⇒ αr β s α−c β −s α−r = αc

It follows that ϕ(β) = 1, a contradiction. Thus, the case (q, v) = (0, 0) cannot occur when u = 1. Suppose (q, v) = (1, 0). Then (5.4.4) becomes αx β y σ z ραa β b αx β y σ z ρ = αa β b which implies that αx β y σ z αa β −b αx β −y σ z = αa β b . Case z = 0: We have αx β y αa β −b αx β −y = αa β b which implies that x = 0 and b = 0. Now ϕ(β ρ ) = ϕ(β −1 ) yields β y ραc β d β y ρ = β −d α−c ⇒ β y αc β −d β −y = β −d α−c

⇒

c = 0.

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Now ϕ(β σ ) = ϕ(β) yields αr β s T β d αr β s T = β d ⇒ αr β s β −d α−r β −s = β d

⇒

d = 0.

It follows that ϕ(β) = 1, a contradiction. Case z = 1: We have αx β y σαa β −b αx β −y = αa β b which implies that αx β y α−a β −b α−x β −y = αa β b . It follows that a = 0 and b = 0 and hence ϕ(α) = 1, a contradiction. Thus, the case (q, v) = (1, 0) cannot occur when u = 1. In other words, when u = 1, v = 1. Suppose v = 1. Case (z, q) = (0, 0): The equality ϕ(σρ)2 = 1 yields (αr β s T αx β y T )2 = 1 ⇒ (αr β s α−x β −y )2 = 1 ⇒ r = x, s = y

⇒

(5.4.5) ϕ(σ) = ϕ(ρ), a contradiction.

Consider the equality ϕ(ρ)2 = 1. This implies that αx β y σ z ρq T v αx β y σ z ρq T v = 1.

(5.4.6)

Case (z, q) = (0, 1): equation (5.4.6) becomes αx β y ρT αx β y ρT = 1 ⇒ αx β y ρα−x β −y β −1 ρ = 1 ⇒α β α x y

−x y+1

β

=1

⇒

(5.4.7) 2y + 1 = 0, a contradiction.

Case (z, q) = (1, 0): equation (5.4.6) becomes αx β y σT αx β y σT = 1 ⇒ αx β y σα−x β −y α−1 σ = 1 x −y

⇒α β α β x y

α=1

⇒

(5.4.8) 2x + 1 = 0, a contradiction.

Case (z, q) = (1, 1): equation (5.4.6) becomes αx β y σρT αx β y σρT = 1 ⇒ αx β y σρα−x β −y α−1 σβ −1 ρ = 1 ⇒ αx β y σα−x β y α−1 σβ = 1 ⇒ αx β y αx β y αβ = 1

⇒

(5.4.9)

2x + 1 = 0, 2y + 1 = 0 a contradiction.

Thus, when u = 1, v = 1. Hence, we conclude that u = 0. Moreover, using the equalities ϕ(σρ)2 = 1 and ϕ(ρ)2 = 1, the equations (5.4.7), (5.4.8), (5.4.9), and (5.4.5) imply that v = 0. Now, f = g = u = v = 0 implies that H is characteristic. Since H = D∞ × D∞ , by Case 6, H has the property R∞ . It follows from Lemma 1.1(3) that G has the property R∞ .

Twisted Conjugacy Classes

137

6. Cases 10, 13, 16 In this section, we show that the wallpaper groups in cases 10 and 16 have the R∞ property but case 13 does not. In each of these cases, the group is a semi-direct product of the form Z2 Zn for n = 3, 4, 6. 6.1. Case 13. This group corresponds to G3 in [8] and has the following presentation G = α, β, t|αβ = βα, αt = α−1 β, β t = α−1 , t3 = 1. Note that G ∼ = Z2 θ Z3 where

−1 −1 θ(t) = 1 0

0 1 . −1 −1

2

and θ(t ) =

= βα and the automorphism Consider the subgroup Z2 = H * ) = α, β|αβ −1 0 α → α−1 ; β → β −1 given by ϕ = . Since αβ = βα, ϕ extends to an 0 −1 automorphism ϕ of G by sending t → t. Moreover, * * ) ) 1 1 0 −1 t · ϕ = θ(t)ϕ = and t2 · ϕ = θ(t2 )ϕ = . −1 0 1 1 Therefore, R(ϕ ) < ∞, R(t · ϕ ) < ∞, and R(t2 · ϕ ) < ∞ since det(1 − ϕ ) = 0, det(1 − t · ϕ ) = 0, and det(1 − t2 · ϕ ) = 0. Since Z3 = G/H is ﬁnite, it follows from Lemma 1.1(2) that R(ϕ) < ∞ and thus G does not have the property R∞ . 6.2. Case 10. This group corresponds to G4 in [8] and has the following presentation G = α, β, t|αβ = βα, αt = β, β t = α−1 , t4 = 1. Note that G ∼ = Z2 θ Z4 where

−1 0 0 −1 2 θ(t ) = θ(t) = 0 −1 1 0 Let ϕ ∈ Aut(G) and be represented by ⎡ a ϕ = ⎣b

0 1 . −1 0

3

and θ(t ) =

⎤ c r d s⎦ . δ γ

(6.2.1)

The equality ϕ(αt ) = ϕ(β) yields αr β s tγ αa β b t t−γ β −s α−r = αc β d tδ . By equating the exponents of t, we have = δ. The equality ϕ(αβ) = ϕ(βα) yields αa β b t αc β d tδ = αc β d tδ αa β b t .

(6.2.2)

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D. Gon¸calves and P. Wong

Case δ = 1 = : Equation (6.2.2) becomes αa β b tαc β d t = αc β d tαa β b t ⇒ αa β b β c α−d = αc β d β a α−b ⇒ a − d = c − b and b + c = a + d

or a = c, b = d.

It follows that ϕ(α) = ϕ(β), a contradiction. Case δ = 2 = : Equation (6.2.2) becomes αa β b t2 αc β d t2 = αc β d t2 αa β b t2 ⇒ αa β b α−c β −d = αc β d α−a β −b ⇒ a − c = c − a and b − d = d − b

or a = c, b = d.

It follows that ϕ(α) = ϕ(β), a contradiction. Case δ = 3 = : Equation (6.2.2) becomes αa β b t3 αc β d t3 = αc β d t3 αa β b t3 ⇒ αa β b β −c αd = αc β d β −a αb ⇒ a + d = c + b and b − c = d − a

or a = c, b = d.

It follows that ϕ(α) = ϕ(β), a contradiction. Thus, δ = 0 = and H = α, β|αβ = βα is characteristic in G. The equality ϕ(αt ) = ϕ(β) yields αr β s tγ αa β b t−γ β −s α−r = αc β d .

(6.2.3)

Case γ = 1: Equation (6.2.3) becomes αr β s tαa β b t−1 β −s α−r = αc β d ⇒ αr β s β a α−b β −s α−r = αc β d . It follows that r − b − r = c and s + a − s = d so that a = d and b = −c. The restriction ϕ = ϕ|H is given by * ) a −b ϕ = . b a Since ϕ is an automorphism, det ϕ = a2 + b2 = 1. Since a, b ∈ Z, it follows that * * ) * ) * ) ) 0 −1 0 1 −1 0 1 0 . , (iv) , (iii) , (ii) ϕ = (i) −1 0 1 0 0 −1 0 1 2 2 For (i), * (iii) and (iv), det(1 − ϕ ) = 0 so R(ϕ ) = ∞. For (ii), t · ϕ = θ(t )ϕ = ) 1 0 so that R(t2 · ϕ ) = ∞. Thus, we can conclude from Lemma 1.1(3) that 0 1 R(ϕ) = ∞.

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Case γ = 2: Equation (6.2.3) becomes αr β s t2 αa β b t−2 β −s α−r = αc β d ⇒ αr β s α−a β −b β −s α−r = αc β d . It follows that r − a − r = c and s − b − s = d so that a = −c and b = −d. The restriction ϕ = ϕ|H is given by * ) a −a ϕ = . b −b This contradicts the fact that ϕ is an automorphism so this case cannot occur. Case γ = 3: Equation (6.2.3) becomes αr β s t3 αa β b t−3 β −s α−r = αc β d ⇒ αr β s β −a αb β −s α−r = αc β d . It follows that r + b − r = c and s − a − s = d so that a = −d and b = c. The restriction ϕ = ϕ|H is given by * ) a b . ϕ = b −a Since ϕ is an automorphism, det ϕ = a2 + b2 = 1. It follows that * * ) * ) * ) ) 0 −1 0 1 −1 0 1 0 . , (iv) , (iii) ϕ = (i) , (ii) −1 0 1 0 0 1 0 −1 For all of these four cases, det(1 − ϕ ) = 0 so R(ϕ ) = ∞. Hence by Lemma 1.1(3) the group G has the property R∞ . 6.3. Case 16. This group corresponds to G6 in [8] and has the following presentation G = α, β, t|αβ = βα, αt = β, β t = α−1 β, t6 = 1. Note that G ∼ = Z2 θ Z6 where

0 −1 −1 −1 −1 0 θ(t2 ) = θ(t3 ) = θ(t) = 1 1 1 0 0 −1

0 1 1 1 and θ(t5 ) = . θ(t4 ) = −1 −1 −1 0 Let ϕ ∈ Aut(G) and be represented by (6.2.1). The equality ϕ(αt ) = ϕ(β) yields αr β s tγ αa β b t t−γ β −s α−r = αc β d tδ . By equating the exponents of t, we have γ + − γ = δ or = δ. Since ϕ(t) is of order 6, it follows that γ = 1 or 5. The equality ϕ(β t ) = ϕ(α−1 β) yields αr β s tγ αc β d tδ t−γ β −s α−r = t− β −b α−a αc β d tδ .

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By equating the exponents of t, we have γ + δ − γ = − + δ Now,

⇒

⎡

a c ϕ = ⎣b d 0 0

δ = 0 ⇒ = 0. ⎤ r s⎦ γ

so that H is characteristic in G. Case γ = 1: the equality ϕ(αt ) = ϕ(β) yields αr β s tαa β b t−1 β −s α−r = αc β d ⇒ αr β s β a α−b β b β −s α−r = αc β d . It follows that r −) b − r = c* and s + a + b − s = d so that b = −c and a + b = d. a −b Now ϕ = ϕ|Z2 = . Since ϕ is an automorphism, a2 + ab + b2 = ±1. b a+b Case γ = 5: the equality ϕ(αt ) = ϕ(β) yields αr β s t5 αa β b t−5 β −s α−r = αc β d ⇒ αr β s αa β −a αb β −s α−r = αc β d . It follows that r +)a + b − r*= c and s − a − s = d so that a = −d and a + b = c. a a+b . Since ϕ is an automorphism, a2 + ab + b2 = ±1. Now ϕ = ϕ|Z2 = b −a If |a| = |b| (say |b| < |a|) and ab < 0 then (a + b)2 < a2 + b2 + ab < a2 + b2 , a contradiction. Similarly, a2 + b2 + ab > 1 if ab > 0. Thus, either ab = 0 or |a| = |b|. It follows that (a, b) ∈ {(1, 0), (0, 1), (1, −1), (−1, 0), (0, −1), (−1, 1)}. When γ = 1, we have * ) * ) -) 0 −1 0 1 0 , , ϕ ∈ 1 0 −1 0 1

*. * ) * ) * ) −1 −1 1 1 0 1 −1 . , , , 1 0 −1 0 −1 −1 1

Note that ϕ = θ(tj ) for j = 0, . . . , 5. It follows that for ϕ , there exists some j such that det(1 − θ(tj )ϕ ) = 0. This implies that R(tj · ϕ ) = ∞ and hence R(ϕ) = ∞ by Lemma 1.1(3). When γ = 5, we have *. * ) * ) * ) * ) * ) -) −1 0 1 0 0 −1 0 1 −1 −1 1 1 . , , , , , ϕ ∈ 1 1 −1 −1 −1 0 1 0 0 1 0 −1 * * ) ) −1 −1 0 −1 If ϕ = then θ(t)ϕ = and thus det(1 − θ(t)ϕ ) = 0 or 0 1 −1 0 R(t · ϕ ) = ∞.

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* * ) 1 0 1 1 then θ(t)ϕ = and thus R(t · ϕ ) = ∞. Likewise, if ϕ = −1 −1 0 −1 For the remaining four cases for ϕ , we proceed as in the previous situation when γ = 1 and conclude that R(tj · ϕ ) = ∞ for some j. Hence, we conclude from Lemma 1.1(3) that R(ϕ) = ∞ and G has the property R∞ . )

7. Cases 11, 12, 14, 15, and 17 In this section, we show that groups in the remaining ﬁve cases all have the property R∞ . In each of these cases, we use an appropriate presentation so that every automorphism can be represented by ⎤ ⎡ a c r x ⎢ b d s y⎥ ⎥ (7.0.1) ϕ=⎢ ⎣ δ γ z⎦ . m n p q First, we will analyze cases 14 and 15. Although the group G3 for case 13 does NOT have the property R∞ and the groups in these two cases are ﬁnite extensions of G3 , it turns out that there are very few automorphisms in these situations. 7.1. Case 14. This group is G13 in [8] and has the following presentation G = α, β, σ, ρ|αβ = βα, ασ = α−1 β, β σ = α−1 , αρ = α, β ρ = αβ −1 , ρ2 = σ 3 = (σρ)2 = 1. Note that G ∼ = Z2 θ (Z3 Z2 ) where Z3 Z2 ∼ = σ, ρ|ρ2 = σ 3 = (σρ)2 = 1. Note also that σρ = σ −1 = σ 2 . Let ϕ ∈ Aut(G). Since ϕ(σ3 ) = 1, αr β s σ γ ρp αr β s σ γ ρp αr β s σ γ ρp = 1. By equating the exponents of ρ, we have 3p = 0 so p = 0. This means that γ = 0 otherwise ϕ(σ) would have inﬁnite order. The equality ϕ(ρ2 ) = 1 implies that (z, q) = (0, 0). Similarly, (z, q) = (1, 0) and (z, q) = (2, 0). Thus, q = 1. Moreover, ϕ(ρ2 ) = 1 implies that αx β y σ z ραx β y σ z ρ = 1 or αx β y σ z αx (αβ −1 )y σ −z = 1

(7.1.1)

When z = 0, (7.1.1) yields y = −2x. When z = 1, (7.1.1) yields x = −2y. When z = 2, (7.1.1) gives x = y. The equality ϕ(ασ ) = ϕ(α−1 β) implies that αr β s σ γ αa β b σ ρm σ −γ β −s α−r = ρ−m σ − β −b α−a αc β d σ δ ρn .

(7.1.2)

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By equating the exponents of ρ, we have m = −m + n or n = 2m = 0 so n = 0. The equality ϕ(β σ ) = ϕ(α−1 ) yields αr β s σ γ αc β d σ δ σ −γ β −s α−r = ρ−m σ − β −b α−a .

(7.1.3)

By equating the exponents of ρ, we have m = 0. The equality ϕ(αρ ) = ϕ(α) implies that αx β y σ z ραa β b σ ρ−1 σ −z β −y α−x = αa β b σ ⇒ αx β y σ z αa (αβ −1 )b σ − σ −z β −y α−x = αa β b σ . By equating the exponents of σ, we have = 0. Now (7.1.3) becomes αr β s σ γ αc β d σ δ σ −γ β −s α−r = β −b α−a . By equating the exponents of σ, we have γ + δ − γ = 0 so δ = 0. Thus, ⎤ ⎡ a c r x ⎢b d s y⎥ ⎥ ϕ=⎢ ⎣0 0 γ z ⎦ . 0 0 0 1 Now, equation (7.1.2) becomes αr β s σ γ αa β b σ −γ β −s α−r = αc−a β d−b . When γ = 1, we have αr β s (α−1 β)a α−b β −s α−r = αc−a β d−b so that c = −b and d = a+b. When γ = 2, we have αr β s β −a (αβ −1 )b β −s α−r = αc−a β d−b so that d = −a and c = a + b. In either case, det ϕ = ±(a2 + ab + b2 ) = ±1. For γ = 2, we have *. * ) * ) * ) * ) * ) -) −1 0 0 −1 −1 −1 1 0 0 1 1 1 . , , , , , ϕ ∈ 1 1 −1 0 0 1 −1 −1 1 0 0 −1 * * ) * ) * ) ) −1 0 −1 −1 1 0 1 1 For ϕ = , det(1 − ϕ ) = 0 so that , or , , 1 1 0 1 −1 −1 0 −1 * * * ) * ) ) ) 0 −1 1 1 0 1 −1 −1 R(ϕ ) = ∞. For ϕ = , θ(σ)ϕ = . , or , or −1 0 0 −1 1 0 0 1 In either case, det(1 − θ(σ)ϕ ) = 0 so that R(θ(σ)ϕ ) = ∞. For γ = 1, we have *. * ) * ) * ) * ) * ) -) −1 −1 0 1 −1 0 1 1 0 −1 1 0 . , , , , , ϕ ∈ 1 0 −1 −1 0 −1 −1 0 1 1 0 1 * * ) * * ) ) ) 0 1 −1 0 1 0 0 −1 For ϕ = , , or , det(1 − ϕ ) = 0. For ϕ = , 0 −1 0 1 1 1 * −1 −1 ) 1 1 . For ϕ = it is easy to see that det(1 − θ(ρ)ϕ ) = 0 where θ(ρ) = 0 −1 * * ) ) −1 −1 1 1 , det(1 − θ(σρ)ϕ ) = 0. , or 1 0 −1 0 It follows from Lemma 1.1(3) that G has the property R∞ .

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7.2. Case 15. This group is G23 in [8] and has the following presentation G = α, β, σ, ρ|αβ = βα, ασ = α−1 β, β σ = α−1 , αρ = β, β ρ = α, ρ2 = σ3 = (σρ)2 = 1. Note that G ∼ = Z2 θ (Z3 Z2 ) where Z3 Z2 ∼ = σ, ρ|ρ2 = σ 3 = (σρ)2 = 1. Note also that σρ = σ −1 = σ 2 . Let ϕ ∈ Aut(G). Since ϕ(σ3 ) = 1, αr β s σ γ ρp αr β s σ γ ρp αr β s σ γ ρp = 1. By equating the exponents of ρ, we have 3p = 0 so p = 0. This means that γ = 0 otherwise ϕ(σ) would have inﬁnite order. The equality ϕ(ρ2 ) = 1 and the fact that σ has order 3 imply that q = 1. The equality ϕ(ασ ) = ϕ(α−1 β) yields αr β s σ γ αa β b σ ρm σ −γ β −s α−r = ρ−m σ − β −b α−a αc β d σ δ ρn . By equating the exponents of ρ, we have m = −m + n or n = 2m = 0 so n = 0. The equality ϕ(β σ ) = ϕ(α−1 ) yields αr β s σ γ αc β d σ δ σ −γ β −s α−r = ρ−m σ − β −b α−a . By equating the exponents of ρ, we have m = 0. Since m = 0, we can equate the exponents of σ to obtain γ + δ − γ = − or δ = −. Now, we have ⎤ ⎡ a c r x ⎢b d s y ⎥ ⎥ ϕ=⎢ ⎣ − γ z ⎦ . 0 0 0 1 Consider ϕ(α3 ) = (αa β b σ )3 . When = 1, 2, it is straightforward to see that ϕ(α3 ) = 1. However, ϕ(α) is of inﬁnite order thus we conclude that = 0. This means that Z2 is characteristic. Following Case 14, we use the equality ϕ(ασ ) = ϕ(α−1 β) to conclude that the restriction ϕ has det ϕ = ±(a2 + ab + b2 ) = ±1 so that *. * ) * ) * ) * ) * ) -) −1 0 0 −1 −1 −1 1 0 0 1 1 1 ϕ ∈ . , , , , , 1 1 −1 0 0 1 −1 −1 1 0 0 −1 It follows that either det(1 − ϕ ) = 0 or det(1 − θ(σ)ϕ ) = 0. We conclude from Lemma 1.1(3) that G has the property R∞ .

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7.3. Case 11. This group corresponds to G14 in [8] and has the following presentation G = α, β, σ, ρ|αβ = βα, ασ = β, β σ = α−1 , αρ = α, β ρ = β −1 , ρ2 = σ 4 = (σρ)2 = 1. Note that G ∼ = G4 θ Z2 where G4 is the wallpaper group in case 10. Let ϕ ∈ Aut(G). The equality ϕ(ασ ) = ϕ(β) yields αr β s σ γ ρp αa β b σ ρm ρ−p σ −γ β −s α−r = αc β d σ δ ρn . Equating the exponents of ρ yields p + m − p = n or m = n. The equality ϕ(β σ ) = ϕ(α−1 ) yields αr β s σ γ ρp αc β d σ δ ρn ρ−p σ −γ β −s α−r = ρ−m σ − β −b α−a . Equating the exponents of ρ yields p + n − p = −m or m = −n. It follows that m = 0 = n. Since ϕ(σ)4 = 1, we have (αr β s σ γ ρp )4 = 1. Suppose p = 1. If γ = 0, (αr β s σ γ ρp )4 = 1 becomes α4r = 1, a contradiction. If γ = 1, then ϕ(σ)4 = (αr β s σραr β s σρ)2 = (αr+s β r+s )2 = 1. It follows that r + s = 0 but this means that ϕ(σ) has order 2, a contradiction. If γ = 2, then (αr β s σ γ ρp )4 = 1 becomes β 4s = 1, again a contradiction. Finally if γ = 3, ϕ(σ)4 = (αr β s σ 3 αr β −s σ −3 )2 = (αr−s β s−r )2 = 1. It follows that r = s and ϕ(σ) has order 2, a contradiction. Moreover, the case r = 0 = s cannot occur for in this case ϕ(σ) = σγ ρ is of order 2, again a contradiction. We conclude that the case when p = 1 cannot occur. Thus, we have m = n = p = 0 which means that G4 is characteristic in G. Since G4 has the property R∞ , it follows that G also has the property. 7.4. Case 12. This group corresponds to G24 in [8] and has the following presentation G = α, β, σ, ρ|αβ = βα, ασ = β, β σ = α−1 , αρ = α, β ρ = β −1 , ρ2 = α, σ 4 = (σρ)2 = 1. Note that G ∼ = G4 θ Z2 where G4 is the wallpaper group in case 10 and Z2 =

t|t2 = 1. Here, the projection G → Z2 is given by sending α, β, σ to 1, ρ → t with kernel G4 and the section Z2 → G is given by t → σρ. Here, the action θ(t) is given by α → β, β → α, σ → σ −1 β −1 . Let ϕ ∈ Aut(G) be given by ⎤ ⎡ a c r x ⎢ b d s y⎥ ⎥ ϕ=⎢ ⎣ δ γ z⎦ m n p q where the columns represent the images of the generators α, β, σ, and t (not ρ).

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Suppose p = 1. The equality ϕ(σ4 ) = 1 yields (αr β s σ γ tαr β s σ γ t)2 = 1.

(7.4.1)

If γ = 0 then (7.4.1) becomes (αr β s tαr β s t)2 = (αr β s β r αs )2 = 1 which implies that r + s = 0. This means that ϕ(σ 2 ) = 1 or ϕ(σ) has order 2, a contradiction. If γ = 1 then (7.4.1) becomes (αr β s σtαr β s σt)2 = (αr β s σβ r αs σ −1 β −1 )2 = (αr β s α−r β s β −1 )2 = (β 2s−1 )2 = 1, a contradiction. Similarly, a straightforward calculation shows that if γ = 2 then (7.4.1) becomes (αr−s+1 β s−r−1 )2 = 1 or r − s + 1 = 0 so that ϕ(σ) has order 2 and that if γ = 3 then (7.4.1) becomes (α2r+1 )2 = 1, a contradiction. Thus, we conclude that p = 0. Now p = 0, ϕ(σ4 ) = 1 yields (αr β s σ γ αr β s σ γ )2 = 1. Since p = 0, γ = 0 or else ϕ(σ) would be of inﬁnite order. If γ = 2, then we have (αr β s σ 2 αr β s σ 2 )2 = (αr β s α−r β −s )2 = 1 so ϕ(σ) has order 2, a contradiction. Thus, γ is either 1 or 3. The equality ϕ(ασ ) = ϕ(β), with p = 0, yields αr β s σ γ αa β b σ tm σ −γ β −s α−r = αc β d σ δ tn . Equating the exponents of t yields m = n. Suppose (m, n) = (1, 1). Note that σt = σ −1 β −1 thus the equality ϕ(σ t ) = −1 −1 ϕ(σ β ) yields αx β y σ z tq αr β s σt−q σ −z β −y α−x = σ −1 β −s α−s t−1 σ −δ β −d α−c . Equating the exponents of t leads to a contradiction so the case (m, n) = (1, 1) cannot occur. Thus, m = n = p = 0 and G4 is characteristic in G. Hence G has property R∞ since G4 does. 7.5. Case 17. This group corresponds to G16 in [8] and has the following presentation G = α, β, σ, ρ|αβ = βα, ασ = β, β σ = α−1 β, αρ = α, β ρ = αβ −1 , ρ2 = σ6 = (σρ)2 = 1. Note that G ∼ = G6 Z2 where G6 is the wallpaper group in case 16 and Z2 =

ρ|ρ2 = 1. Let ϕ ∈ Aut(G) be given by (7.0.1). The equality ϕ(β σ ) = ϕ(α−1 β) yields αr β s σ γ ρp αc β d σ δ ρn ρ−p σ −γ β −s α−r = ρ−m σ − β −b α−a αc β d σ δ ρn .

(7.5.1)

Equating exponents of ρ yields p + n − p = −m + n which implies that m = 0. The equality ϕ(ασ ) = ϕ(β) yields αr β s σ γ ρp αa β b σ ρ−p σ −γ β −s α−r = αc β d σ δ ρn . Equating exponents of ρ yields p − p = n which implies that n = 0.

(7.5.2)

146

D. Gon¸calves and P. Wong Suppose p = 1. The equality ϕ(σ 6 ) = 1 becomes (αr β s σ γ ρ)6 = (αr β s σ γ ραr β s σ γ ρ)3 = 1

or

(αr β s σ γ αr (αβ −1 )s σ −γ )3 = 1. (7.5.3) γ Note that if r = 0 = s then (7.5.3) shows that ϕ(σ) = σ ρ has order 2, a contradiction. We assume that (r, s) = (0, 0). Caseγ = 0: (7.5.3) reduces to (α2r+s )3 = 1. That means 2r + s = 0 and so ϕ(σ) has order 2, a contradiction. Caseγ = 1: (7.5.3) reduces to (αr+s β r+s )3 = 1. That means r + s = 0 and so ϕ(σ) has order 2, a contradiction. Caseγ = 2: (7.5.3) reduces to (β r+2s )3 = 1. That means r + 2s = 0 and so ϕ(σ) has order 2, a contradiction. Caseγ = 3: (7.5.3) reduces to (α−s β 2s )3 = 1. That means s = 0 and so ϕ(σ) has order 2, a contradiction. Caseγ = 4: (7.5.3) reduces to (αr−s β r+s )3 = 1. That means r = s = 0, a contradiction. Caseγ = 0: (7.5.3) reduces to (α2r β −r )3 = 1. That means r = 0 and so ϕ(σ) has order 2, a contradiction. Hence, we conclude that the case when p = 1 cannot occur and so p = 0. Since m = 0 = n, the subgroup G6 is characteristic. Since G6 has the property R∞ , so does G.

8. Concluding remarks Given a short exact sequence of groups 1 → Zn → G → F → 1 where F is a ﬁnite group, F acts on the kernel by conjugation in G and the action is given by θ : F → GLn (Z). If this action is injective and n = 2, then the isomorphism classes of such groups G are precisely the 17 wallpaper groups. It is natural to investigate the R∞ property for such extensions with injective actions in higher dimensions. There are some partial results in this direction based upon by our knowledge of the cases n = 2. However the complete solution of the problems is much more complex and we intend to investigate this problem in the sequel.

References [1] H.S.M. Coxeter and W.O.J. Moser, Generators and relations for discrete groups. Springer-Verlag, Berlin-G¨ottingen-Heidelberg, 1957. viii + 155 pp. [2] W. Dicks and M. Dunwoody, Groups acting on graphs. Cambridge Studies in Advanced Mathematics, 17. Cambridge University Press, Cambridge, 1989. xvi+283 pp. [3] A.L. Fel’shtyn, The Reidemeister number of any automorphism of a Gromov hyperbolic group is inﬁnite, Zapiski Nauchnych Seminarov POMI 279 (2001), 229–241.

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[4] D. Gon¸calves and P. Wong, Twisted conjugacy classes in exponential growth groups, Bull. London Math. Soc. 35 (2003), 261–268. [5] D. Gon¸calves and P. Wong, Twisted conjugacy classes for nilpotent groups, J. Reine Angew. Math. 633 (2009), 11–27. [6] B. Jiang, “Lectures on Nielsen Fixed Point Theory,” Contemporary Mathematics vol. 14, Amer. Math. Soc., Providence, Rhode Island, 1983. [7] G. Levitt and M. Lustig, Most automorphisms of a hyperbolic group have very simple ´ Norm. Sup. 33 (2000), 507–517. dynamics, Ann. Scient. Ec. [8] R. Lyndon, Groups and Geometry. LMN Lecture Note Series, 101, Cambridge University press, 1987. Daciberg Gon¸calves Dept. de Matem´ atica – IME – USP Caixa Postal 66.281 – CEP 05314-970 S˜ ao Paulo – SP, Brasil FAX: 55-11-30916183 e-mail: [email protected] Peter Wong Department of Mathematics Bates College Lewiston, ME 04240, USA FAX: 1-207-7868331 e-mail: [email protected]

Combinatorial and Geometric Group Theory Trends in Mathematics, 149–169 c 2010 Springer Basel AG

Solving Random Equations in Garside Groups Using Length Functions Martin Hock and Boaz Tsaban Abstract. We give a systematic exposition of memory-length algorithms for solving equations in noncommutative groups. This exposition clariﬁes some points untouched in earlier expositions. We then focus on the main ingredient in these attacks: Length functions. After a self-contained introduction to Garside groups, we describe length functions induced by the greedy normal form and by the rational normal form in these groups, and compare their worst-case performances. Our main concern is Artin’s braid groups, with their two known Garside presentations, due to Artin and due to Birman-Ko-Lee (BKL). We show that in B3 equipped with the BKL presentation, the (eﬃciently computable) rational normal form of each element is a geodesic, i.e., is a representative of minimal length for that element. (For Artin’s presentation of B3 , Berger supplied in 1994 a method to obtain geodesic representatives in B3 .) For arbitrary BN , ﬁnding the geodesic length of an element is NP-hard, by a 1991 result of by Paterson and Razborov. We show that a good estimation of the geodesic length of an element of BN in Artin’s presentation is measuring the length of its rational form in the BKL presentation. This is proved theoretically for the worst case, and experimental evidence is provided for the generic case. Mathematics Subject Classiﬁcation (2000). 05E15, 94A60. Keywords. Random equations, Garside groups, length functions, braid group, Artin presentation, Birman-Ko-Lee presentation, minimal length, geodesics.

1. Solving random equations All groups considered in this paper are multiplicative noncommutative groups, with an eﬃciently solvable word problem, that is, there is an eﬃcient algorithm for deciding whether two given (ﬁnite products of) elements in the group are equal as elements of the group. Throughout this paper, G denotes such a group. The second author was partially supported by the Koshland Center for Basic Research.

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Problems involving solutions of equations in groups have a long history, and are nowadays also explored towards applications in public-key cryptography [14]. We mention some of the more elegant problems of this type. Problem 1 (Conjugacy Search). Given conjugate a, b ∈ G, ﬁnd x ∈ G such that b = xax−1 . Problem 2 (Root Search). Given a ∈ G, ﬁnd x ∈ G such that a = x2 , provided that such x exists. Problem 3 (Decomposition Search). Let H be a proper subgroup of G. Given a, b ∈ G, ﬁnd x, y ∈ H such that b = xay, provided that there exist such x, y. We will discuss the meaning of the terms “given” and “ﬁnd”, appearing in Problems 1–3, later. Problems 1–3, as well as many additional ones, can be stated generally as follows. By free-group word w(t1 , . . . , tk ) we mean a product of variables ti11 · ti22 · · · · · tinn for any choice of a positive integer n and elements i1 , . . . , in ∈ {1, . . . , k} and 1 , . . . , n ∈ {1, −1}, such that no cancellation is possible, that is, for each j = 1, . . . , n, if ij = ij+1 , then j = −j+1 . Problem 4 (Solution Search). Fix H1 , . . . , Hk ≤ G and a free-group word w(t1 , . . . , tk+n ). Given parameters p1 , . . . , pn ∈ G and an element c ∈ G, ﬁnd x1 ∈ H1 , . . . , xk ∈ Hk such that c = w(x1 , . . . , xk , p1 , . . . , pn ), provided that there exist such x1 , . . . , xk . Problem 4 deals with the solution of a single solvable equation (with parameters). It can also be stated for systems of several equations. The algorithms proposed here easily generalize to cover this case, cf. [10]. 1.1. Making the problems meaningful It suﬃces to discuss Problem 4. First, all given information must be coded in some compact form. For example, the subgroups H1 , . . . , Hk of G may be described by lists of generators and relations, all (the list, the generators, and the relations) of manageable length. Second, the problem may require that it be possible to ﬁnd a solution for each possible instance of the problem, or for a certain portion of the instances. Already in the case of free groups, the problem of solving equations in this sense is extremely diﬃcult. For example, the problem of solving quadratic equations over free groups is known to be NP-hard. Alternatively, the instances of the problem may be chosen according to a certain distribution D, and we may require that a solution can be found with a high-enough probability (a probabilistic model ). Finally, by “ﬁnd” we mean “ﬁnd eﬃciently”, i.e., use an algorithm with a feasible running time. Otherwise, in most cases of interest the problems are solvable. E.g., if G is a ﬁnitely generated group with solvable word problem, then we can solve Problem 4 by enumerating Gk recursively, and trying all possible solutions

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until one is found. This algorithm always succeeds in a ﬁnite running time, but usually this running time is infeasible. In this discussion, all quantitative terms (compact, eﬃcient, signiﬁcant, etc.) have two natural interpretations: Concrete (e.g., of size less than 1GB) or asymptotic (e.g., polynomial in the size of the input). 1.2. The probabilistic model With an eye towards applications, we will always use the probabilistic version of the problems, where we wish to ﬁnd (eﬃciently) a solution with a signiﬁcant probability, provided that the instances of the problem are chosen according to a certain known distribution D. More precisely, in Problem 4 we ﬁx a distribution D on Gk+n such that for each (x1 , . . . , xk , p1 , . . . , pn ) in the support of D, we have that x1 ∈ H1 , . . . , xk ∈ Hk . An instance of the problem is generated as follows: A secret tuple (x1 , . . . , xk , p1 , . . . , pn ) ∈ Gk+n is chosen according to the distribution D, and we are given p1 , . . . , pn and an element c ∈ G equal to w(x1 , . . . , xk , p1 , . . . , pn ) in G. We must then search for elements x ˜1 ∈ H1 , . . . , x ˜k ∈ Hk such that with a signiﬁcant probability, c = w(˜ x1 , . . . , x ˜k , p1 , . . . , pn ) in G. By peeling oﬀ known parameters on the left of the given word w(x1 , . . . , xk , p1 , . . . , pn ), we may assume that it begins with a variable xi (possibly inverted). If we are able to ﬁnd xi (with a signiﬁcant probability), we can treat it as a parameter henceforth, and proceed to the next leading variable after peeling oﬀ all parameters on the left. Continuing in this manner, we ﬁnd suggestions for all variables, and can check whether we obtained a solution. Thus, it is natural to consider the following problem. Problem 5 (Leading-Variable Search). Fix H1 , . . . , Hk ≤ G and a free-group word t1 · w(t1 , . . . , tk+n ). Given parameters p1 , . . . , pn ∈ G and an element c = x1 · w(x1 , . . . , xk , p1 , . . . , pn ) ∈ G such that x1 ∈ H1 , . . . , xk ∈ Hk , ﬁnd x ˜1 ∈ H1 , such that there are x ˜2 ∈ H 2 , . . . , x ˜k ∈ Hk with c = x ˜1 · w(˜ x1 , . . . , x ˜k , p1 , . . . , pn ). Clearly, any algorithm solving Problem 4 also solves Problem 5, with at least the same probability of success. On the other hand, an algorithm for Problem 5 can be iterated, as explained above, to obtain a solution for Problem 4 (with a smaller probability of success, which also depends on its performance on the induced distributions along the iteration). 1.3. Decision problems All mentioned problems also have a decision version. For example, the Congugacy Problem is: Given a, b ∈ G, are they conjugate? If we only consider algorithms with bounded running time, then a solution to the search version also implies a solution to the decision version, in the following sense. Assume that A is an algorithm searching for solutions of equations of a certain type (e.g., b = xax−1 ), and that its running time is bounded, say by a certain function of the length of its input. We deﬁne a decision algorithm A with running

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time bounded by the same function: Given an instance of the equation to be checked, run A on this instance until the running time reaches its bound, and then terminate it if it did not terminate already. If a solution was found, the decision of A is Yes. Otherwise, it is No. Assume that the instances of the equation are distributed according to some distribution E. This induces a distribution D on the solvable equations, by conditioning that the chosen equation be solvable. Let p be the probability that A ﬁnds a solution to (necessarily, solvable) equations distributed according to D. For each speciﬁc instance of the equation, A is correct in probability at least p: If this instance has a solution, it will be found by A in probability p, in which case A decides “Yes”. And if this instance has no solution, then in probability 1, A will not ﬁnd a solution (because there is none), and A decides “No”. This can also be viewed as follows: Let q = 1 − p. The probability that A comes up with a wrong answer is: P (Wrong decision) = = P (Decision = Yes | Solution) · P (Solution) + + P (Decision = No | ∃Solution) · P (∃Solution) = = 0 · P (Solution) + q · P (∃Solution) = = q · P (∃Solution). In particular, this probability is at most q, and the worst case is when P (∃Solution) is 1, in which the distribution may be assumed to be supported by solvable instances, and we are actually in the search version of the problem. This justiﬁes, to some extent, restricting attention to search problems when working in the probabilistic model, with algorithms of bounded running time.

2. The memory-length approach The potential usefulness of length functions for solving the conjugacy search problem was identiﬁed in [11]. In [9, 10], it was pointed out that this approach can be used to solve arbitrary (systems of) equations. Let H ≤ G be generated by elements a1 , . . . , am of G. Assume that an instance x · w(x, x2 , . . . , xk , p1 , . . . , pn ) of Problem 5 is chosen according to a certain distribution D, with H1 = H, and we are given c which is equal to it in G. Let w = w(x, x2 , . . . , xk , p1 , . . . , pn ). Let A = {a1 , . . . , am }±1 . Assume that the shortest expression of x as a product of elements of A has length n. Let COR(x) be the set of all a ∈ A which appear ﬁrst in an expression of x as a product of n generators, i.e., {a ∈ A : x ∈G a·An−1 }. For each a ∈ COR(x), a−1 x has an expression of length n − 1, whereas for a∈ / COR(x), a−1 x may in general not have an expression shorter than n+1. In particular, we expect a−1 x to be “shorter” when a ∈ COR(x) than when a ∈ / COR(x). Heuristically, this expectation is extended to xw.

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Often, we cannot compute the length of a shortest expression of a group element, and we only assume that we have an eﬃciently computable function : G → R≥0 , which approximates the above situation, i.e., such that (abw) tends to be greater than (w) for w ∈ G, a, b ∈ {a1 , . . . , am }±1 . By standard arguments, we may for convenience assume that n is known [10, 16].1 One may then try all a ∈ A, and pick one with (a−1 xw) minimal. Hopefully, a ∈ COR(x), and we can continue with the peeled-oﬀ element a−1 xw. After n steps, we hopefully have (a shortest expression for) x. In cases of interest this approach does not work as stated [9], and the following improvement was proposed in [10]. 2.1. The memory-length algorithm Using the above-mentioned notation, the algorithm generates an ordered list of M sequences of length n, with the aim that with a signiﬁcant probability, a sequence ((j1 , 1 ), (j2 , 2 ), . . . , (jn , n )), such that x = aj11 aj22 . . . ajnn in G, appears in the list, and tends to be among its ﬁrst few members. It consists of the following steps: − Step 1. For each j = 1, . . . , m and each ∈ {1, −1}, compute a− j c = aj xy, and − give (j, ) the score (aj c). Keep in memory the M elements (j, ) with the best (=lowest) scores.

Steps s = 2, 3, . . . , n. For each sequence ((j1 , 1 ), . . . , (js−1 , s−1 )) out of the M sequences stored in the memory, each js = 1, . . . , m, and each s ∈ {1, −1}, compute −

−

−1 −s −1 s−1 s−1 s (a− js (ajs−1 · · · aj1 c)) = (ajs ajs−1 · · · aj1 xy),

and assign this score to the sequence ((j1 , 1 ), . . . , (js , s )). Keep in memory only the M sequences with the best scores. The algorithm terminates after n steps, with M proposals for ((j1 , 1 ), (j2 , 2 ), . . . , (jn , n )). It is not diﬃcult to see that the complexity of this algorithm is n(n + 4m + 1)M/2 group operations and evaluations of . It is interesting to note that this algorithm may also be useful for solving the following. Problem 6 ((Shortest) Subgroup Membership Search). Given a1 , . . . , am ∈ G and x ∈ a1 , . . . , am , ﬁnd a (shortest possible) expression of x as a product of elements from the set {a1 , . . . , am }±1 . 1 This has a computational cost, so we cannot assume that we know the lengths of shortest expressions of many elements.

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2.2. Suﬃciency for the general problem Assume that the algorithm succeeds, with a signiﬁcant probability, to have the leading element x in the ﬁnal list. Then we have the following. If there is only one unknown variable in the equation (e.g., Problems 1–3), then we can check (in running time M ) all elements in the list and ﬁnd one which is a solution to the problem. In the general case (Problem 4) there are several unknown variables, and we can iterate the algorithm by checking each suggestion in the list. The overall complexity is in principle M k . However, the suggestions for each variable are ordered more or less according to their likelihood, and it suﬃces to check, for some N " M , the N most likely solutions. This reduces the complexity to N k , or more precisely to N1 · N2 · · · Nk , where Nk is the number of elements required at the kth step, and it is likely that Ni+1 " Ni for each i. 2.3. Improvements Certain simple modiﬁcations in the memory-length algorithm increase its success rates. We refer the reader to [16] for details. 2.4. The length function For this algorithm to be meaningful and useful, one must have a good and efﬁciently computable length function on the group G. Our introduction of the memory-length algorithm suggests a natural model for comparing length functions for appropriateness to this method. We explore this below, after introducing a new proposal for a length function on the braid group. The braid group is, thus far, the most popular in applications related to cryptography [14]. Most of these cryptographic applications give rise to an equation, whose solution would imply the insecurity of the application. Thus, it is natural to look for good length functions on this group. See [14] for more details.

3. Excursion: Garside groups We are going to consider two Garside structures on the braid group (to be deﬁned). This section is an essentially self-contained introduction to Garside groups, and may be skipped by readers who are familiar with this concept, and by readers who do not insist on understanding all details of this paper. Garside groups were introduced by Dehornoy and Paris [6], and later in a more general form by Dehornoy [5]. We treat the latter, more general case. All unproved assertions, as well as most of the proved ones, are from [6]. 3.1. Garside monoids and groups Let M be a monoid with cancellation. x ∈ M is an atom if x = 1, and x = ab for a, b ∈ M implies a = 1 or b = 1. M is atomic if M is generated by its atoms, and for each a ∈ M , the maximum number of atoms in an expression of a as a product of atoms, denoted #a#, exists. It follows that #ab# ≥ #a# + #b# for all a, b ∈ M .

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In particular, as 1 = 1 · 1, we have that #1# ≥ #1# + #1#, and thus #1# = 0. For a = 1, #a# > 0. Let M be an atomic monoid. For a, b ∈ M , a is a left divisor of b if there is c ∈ M such that ac = b. Similarly, a is a right divisor of b if there is c ∈ M such that ca = b. a ∈ M is a Garside element of M if its left divisors and right divisors coincide, and include all atoms of M . M is a Garside monoid if it is atomic, has a Garside element, and for all a, b ∈ M , a greatest common divisor a ∧ b and a least common multiple a ∨ b of a and b exist in M , both with respect to left divisibility. For a, b ∈ M , the complement a \ b is the unique c ∈ M such that ac = a ∨ b. The closure of the set of atoms under the operations of complement and least common multiple is the set S of simple elements of M . The least common multiple of all elements of S, if it exists (e.g., if M is ﬁnitely generated), is called the fundamental element of M and denoted δ. δ, if it exists, is the least Garside element of M . G is a Garside group if it is the group of fractions of a Garside monoid M . In this case, the elements of M are called the positive elements of G. In the remainder of this section, M is a Garside group with a fundamental element δ, and G is the Garside group of fractions of M . 3.2. Greedy normal form For x ∈ M with x = 1, the simple element δ ∧ x = 1. Deﬁne ∂(x) = (δ ∧ x)−1 x. Then ∂(x) ∈ M , and as x = (δ ∧ x)∂(x), #x# ≥ #δ ∧ x# + #∂(x)# > #∂(x)#. Deﬁne simple elements s1 , s2 , . . . , as follows. Set x1 = x, and for each i = 1, . . . , r, let si = δ ∧ xi , and xi+1 = ∂(xi ). #x# = #x1 # > #x2 # > · · · ≥ 0, and thus there is a minimal n such that xn+1 = 1. x = s1 · · · sn . Let k ≥ 0 be maximal with si = δ, and deﬁne pi = sk+i , i = 1, . . . , r, r = n − k. The expression x = δ k p1 · · · pr is called the greedy normal form of x. Consider now x ∈ G\M . If x = δ k s and s ∈ M , then k < 0. Take the maximal integer k such that x = δ k s for some s ∈ M . Fix such s, and let δ 0 p1 · · · pr = p1 · · · pr be the greedy normal form of s. The greedy normal form of x is then again deﬁned to be δ k p1 · · · pr . −1 By the construction, we have that pi+1 ∧ p−1 i δ = (pi+1 · · · pr ∧ δ) ∧ pi δ = −1 −1 pi+1 · · · pr ∧ (δ ∧ pi δ) = xi+1 ∧ pi δ = 1 for all i = 1, . . . , r − 1, and that pr = 1. We say in such cases that the sequence p1 , . . . , pr is left-weighted. 3.3. Rational normal form Following Thurston [7, Chapter 9], Dehornoy and Paris deﬁne the rational normal form 2 of an element x ∈ G. To this end, we need the following. Theorem 7 (Dehornoy-Paris [6]). For each x ∈ G, there is a unique pair (u, v) in M × M such that x = u−1 v and u ∧ v = 1. 2 Also

called mixed or symmetric normal form.

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Let x ∈ G, and let u, v ∈ M be as in Theorem 7. Let s1 · · · sk , p1 · · · pl be the greedy normal form of u, v, respectively. The rational normal form of x is the expression x = (s1 · · · sk )−1 (p1 · · · pl ). All si , pj are simple, s1 ∧ p1 = 1, and the sequences s1 , . . . , sk and p1 , . . . , pl are both left-weighted. (The special cases where k = 0 or l = 0 are also allowed.) For each a ∈ G, deﬁne τ (a) = aδ = δ −1 aδ. τ is an inner automorphism of G, n and its nth iterate at a is τ n (a) = aδ . τ maps simple elements to simple elements: For each simple s, let p be such that sp = δ. Then p is simple, and thus there is a simple q with pq = δ. Then sδ = spq = δq, and thus sδ = q is simple. In particular, M is invariant under τ . Any automorphism of G mapping positive elements to positive elements, maps atoms to atoms. It follows that τ is a permutation of the atoms of M . One can obtain the rational normal form from the greedy normal form. To see this, we use the following. Lemma 8. If s, p are simple and sp is left-weighted, then so are sδ pδ and sδ ±1

±1

±1

±1

Proof. If ac = b are all positive, then aδ cδ = (ac)δ = bδ , and cδ Thus, τ ±1 both map left divisors to left divisors, and therefore (a ∧ b)δ

±1

= aδ

±1

∧ bδ

−1

±1

pδ

−1

.

∈ M.

±1

for all a, b ∈ M . Now, assume that sp is left-weighted. Then (sδ

±1

)−1 δ ∧ pδ

showing that sδ

±1

pδ

±1

±1

= (s−1 δ)δ

±1

∧ pδ

±1

= (s−1 δ ∧ p)δ

±1

= 1δ

±1

= 1,

is left-weighted.

Proposition 9. If s, p are simple and sp is left-weighted, then so are ((pδ )−1 δ)((sδ k

k+1

)−1 δ),

for all integer k. Proof. Assume that sp is left-weighted. Then so is (p−1 δ)((sδ )−1 δ): (p−1 δ)−1 δ ∧ ((sδ )−1 δ) = pδ ∧ (s−1 δ)δ = (p ∧ (s−1 δ))δ = 1δ = 1. By Lemma 8, ((pδ )−1 δ)((sδ k

k+1

)−1 δ) = ((p−1 δ)((sδ )−1 δ))δ is also left-weighted. k

Let δ k p1 · · · pr be the greedy normal form of x. Consider three possible cases. Case 1: k ≥ 0. Then δ k p1 · · · pr is already a rational normal form (with a trivial negative part).

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Case 2: k = −m < 0 and m ≥ r. By deﬁnition, δ−n a = aδ δ −n for all a and all n. Using this, we have that n

δ −m p1 · · · pr = δ −1 pδ1 · δ −1 pδ2 · · · · · δ −1 pδr · δ −(m−r) −1 m−r m−2 m−1 = δ m−r · (pδr )−1 δ · · · · · (pδ2 )−1 δ · (pδ1 )−1 δ . m−1

m−2

m−r

By Proposition 9, the last inverted expression is left-weighted, and thus we have a rational form, with a trivial positive part. Case 3: k = −m < 0 and m < r. In the same manner, we have that · δ −1 pδ2 · · · · · δ −1 pm · pm+1 · · · · · pr δ −m p1 · · · pr = δ −1 pδ1 −1 m−1 δm−2 −1 = p−1 ) δ · (pδ1 )−1 δ (pm+1 · · · · · pr ), m δ · · · · · (p2 m−1

m−2

By Proposition 9, each of the bracketed expressions is left-weighted. Thus, this expression is in rational normal form.

4. Several length functions on Garside groups Let M be a Garside monoid with fundamental element δ, and G be its group of quotients. Assumption 10. We assume that for each simple s ∈ M , the minimal length (s) of an expression of s as a product of atoms can be eﬃciently computed. There is always an algorithm for computing (s): Enumerate all words of length 1, 2, 3, . . . , until one equal to s is found. The running time is bounded by k (a) ≤ k a , where k is the number of atoms. But this is in general infeasible. When Assumption 10 fails, one may use in applications an estimation of instead of the true function. Fortunately, in the speciﬁc monoids in which we are interested, all relations are length-preserving, and thus (s) is just the length of any expression of s as a product of atoms. Thus, Assumption 10 is true in our applications. + Example 11 (Artin’s presentation of BN ). Consider the monoid BN generated by σ1 , . . . , σN −1 , subject to the relations

σi σi+1 σi σi σj

= σi+1 σi σi+1 ; = σj σi when |i − j| > 1.

+ The quotient group of this monoid is the braid group BN on N strings. BN is a Garside monoid with atoms σ1 , . . . , σN −1 , and fundamental element

δ = (σ1 · · · σN −1 )(σ1 · · · σN −2 ) · · · (σ1 σ2 )σ1 . The positive elements of BN are the words in σ1 , . . . , σN −1 not involving inverses of generators. As the relations are length preserving, all expressions of a positive

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element as a product of atoms have the same length. Thus, for a ∈ M , #a# is the length of a (any) presentation of a. Elements of BN can be identiﬁed with braids having N strings, where each generator σi performs a half-twist on the ith and i + 1st strings. This way, δ is a half-twist of the full set of strings. The simple elements correspond to positive braids in which any two strings cross at most once. A simple element is described uniquely by the permutation it induces on the strings, and every permutation of the N strings corresponds to a simple element. Example 12 (BKL presentation of BN ). Generalizing the geometric interpretation in Example 11 to allow half-twists of the ith and the jth string for arbitrary i, j, Birman, Ko, and Lee [3] introduced the following presentation of the braid group BN . The monoid BKL+ N is generated by at,s , 1 ≤ s < t ≤ N , subject to the relations at,s ar,q = ar,q at,s if (t − r)(t − q)(s − r)(s − q) > 0; at,s as,r = at,r at,s = as,r at,r if t > s > r. Also here, the relations are length preserving, and thus the norm is equal to the number of atoms in any expression of the element. This monoid also has the braid group BN as its quotient group. In terms of Artin’s presentation (Example 11), the Birman-Ko-Lee (BKL) generators can be expressed by −1 −1 at,s = (σt−1 · · · σs+1 )σs (σs+1 · · · σt−1 ). BKL+ n is a Garside monoid with fundamental element δ = an,n−1 an−1,n−2 · · · a2,1 . Here too, a simple element is described uniquely by the permutation it induces on the strings. However, not every permutation of the n strings corresponds to a simple element. Deﬁnition 13. Let M be a Garside monoid with Garside group G, and let x ∈ G. 1. (x), the minimal length of x, is the minimal length of an expression of x as a product of elements of A±1 , where A is the set of atoms of M . 2. G (x), the greedy length of an x, is the sum of the minimal lengths of all simple elements (including the inverted ones) in the greedy normal form of x. Similarly: 3. R (x), the rational length of x, is the sum of the minimal lengths of all simple elements (including the inverted ones) in the rational normal form of x. Speciﬁcally, if the greedy normal form of x is δ k s1 · · · sr , then G (x) = k · (δ) + (s1 ) + · · · + (sr ), and if the rational normal form of length of x is (s1 . . . sk )−1 p1 . . . pl , then R (x) = (s1 ) + · · · + (sk ) + (p1 ) + · · · + (pl ). Proposition 14. For each a ∈ M , (aδ ) = (a).

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Proof. Let n = (a), and a = a1 · · · an with a1 , . . . , an atoms. Then aδ = aδ1 · · · aδn . As conjugation by δ moves atoms to atoms, (aδ ) ≤ n = (a). Similarly, if m = δ −1

−1

(aδ ) and aδ = b1 · · · bm with b1 , . . . , bm atoms, then a = aδ = bδ1 δ as conjugation by δ moves atoms to atoms, (a) ≤ m = (a ).

−1

· · · bδm , and

The presentation in the previous section of the rational normal form in terms of the greedy normal form gives the following. Corollary 15. The rational length of an element with greedy normal form δ −m s1 · · · sr , where 0 < m ≤ r, is −1 (s−1 1 δ) + · · · + (sm δ) + (sm+1 ) + · · · + (sr ),

and similarly for the cases where m ≤ 0 or 0 < r < m. Corollary 16. If the relations of M are length-preserving, then the rational length of an element with greedy normal form δ k s1 · · · sr can be obtained by removing min(r,k) (si ) from its greedy normal length. 2 i=1 Proof. If the relations of M are length-preserving, we have that (ab) = (a) + (b) for all a, b ∈ M , and thus for simple s, (δ) = (s) + (s−1δ), that is, (s−1 δ) = (δ) − (s). This shows, in particular, that the length function considered in [9, 10] in the case of the Artin presentation of BN is in fact the rational length for the Artin presentation of BN . This was ﬁrst pointed out to us by Dehornoy. 4.1. Quasi-geodesics in Garside groups Even when the relations are length-preserving, it is generally not the case that an eﬃcient algorithm for computing the minimal length (x) is available. Even if the monoid relations are length-preserving, ﬁnding (x) for x not in the monoid (nor in its inverse) may be a diﬃcult task. Indeed, assuming P = N P , there is no polynomial-time algorithm computing (x) with respect to the Artin presentation of BN , for arbitrary N and x ∈ BN [15]. Fortunately, in Garside groups (x) can be approximated. For simplicity, we treat the case of length-preserving relations, so that is easy to compute on positive elements. Theorem 17. Let M be a Garside monoid with length preserving relations and fundamental element δ, and let G be its fractions group. For each x ∈ G: 1. If x ∈ M , then G (x) = R (x) = (x). 2. If x ∈ M −1 , then R (x) = (x). 3. (x) ≤ R (x) ≤ G (x) ≤ (2(δ) − 1)(x). 4. R (x) ≤ ((δ) − 1)(x). Moreover, these bounds in (3) cannot be improved. Proof. (1) For x ∈ M , each normal form gives some positive presentation of x, and thus the corresponding length is the same as the minimal length.

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M. Hock and B. Tsaban (2) Fix x ∈ M −1 . Then R (x) = R (x−1 ), and by (1), R (x−1 ) = (x−1 ) =

(x). (3) The ﬁrst inequality is clear. The second follows from Corollary 16. We prove the third. Let x = a11 · · · amm (1) with m = (x), a1 , . . . , am atoms, and 1 , . . . , m ∈ {1, −1}. For each atom a, let a ¯ be the simple element such that a ¯a = δ. Then a−1 = δ −1 a ¯. Rewrite each negative atom in the equation 1 in this form, and move all occurrences of δ −1 to −1 the left, using the relation aδ −1 = δ −1 aδ . Let n = |{i : i = −1}|. We obtain a presentation x = δ −n b1 · · · bm , with each bi being (up to an application of τ an integer number of times, which preserves length by Proposition 14) ai if i = 1, and a ¯i otherwise. In particular, (bi ) = 1 if i = 1, and (¯ ai ) = (δ) − 1 otherwise. Let δ k s1 · · · sj be the left-weighted form of b1 · · · bm . Then the greedy normal form of x is δ −n+k s1 · · · sj , which cannot be longer than δ −n δ k s1 · · · sj . As expressions of positive elements all have the same length, the length of δ k s1 · · · sj is exactly that of b1 · · · bm . Thus, G (x)

≤ =

n(δ) + (b1 · · · bm ) = n(δ) + (b1 · · · bm ) n(δ) + n((δ) − 1) + (m − n)

=

n(2(δ) − 2) + m ≤ (2(δ) − 1)m,

as n ≤ m. (4) This can be proved as in the proof of (3). Alternatively, one can use Charney’s Theorem [4], extended to general Garside groups by Dehornoy and Paris [6], that the number of simple elements in the rational normal form is minimal amongst presentations of x as a product of simple elements (possibly inverted): If x ∈ M ±1 , we can use (1) or (2) and there is nothing to prove. Otherwise, let x = a11 · · · amm be a minimal presentation of x. In particular each ai 1 is a (possibly inversed) simple element. Thus, the number n of simple elements in the rational form of x is at most m. As x ∈ / M ±1 , no simple element in the rational form of x is δ. It follows that R (x) ≤ ((δ) − 1)m. 3

(1) shows that the lower bounds cannot be improved. To see that the upper bounds in (3) cannot be improved, consider G (a−m ) for m positive and an atom a. The following corollary of Theorem 17 is of special interest. In 1994, Berger supplied an eﬃcient method to compute a minimal length representative of an element of B3 , in terms of Artin generators [2]. We show that the same is true 3 The step before last is added to emphasize that for random words, the upper bound is far from being optimal. Indeed, in this case we have n ≈ m/2, which gives roughly half of the mentioned bound. There is an elbow room for improvements in the random case.

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for the BKL presentation. Indeed, a minimal length representative for the BKL presentation is supplied by the rational normal form. Corollary 18. Consider the BKL presentation of B3 . For each x ∈ B3 , R (x) = (x). Proof. Here, (δ) = 2. By Theorem 17, (x) ≤ R (x) ≤ ((δ) − 1)(x) = (x).

Remark 19. Let M be a Garside monoid, and G be its fractions group. Dehornoy and Paris [6] proved that for each x ∈ G, there is a unique pair (u, v) ∈ M 2 , such that x = u−1 v. It follows that for each braid x, the rational normal form of x belongs to BN with the smallest possible N . In fact, if we deﬁne the support of a braid as the set of strands that cross in every braid representative, then the rational normal form of x detects its support. This is another reason why rational normal forms approximate the minimal length. Remark 20. We do not know whether the upper bound in (4) of Theorem 17 can be improved. At ﬁrst it seems that for positive m and distinct non-commuting atoms a, b, R (am b−m ) = ((δ) − 1)(am b−m ), but this is not the case: Consider σ22 σ1−2 in the Artin presentation of BN . Its rational normal form in B3 (and thus by Remark 19 in BN for all N ) is (σ1−1 σ2−1 ) · (σ2−1 σ1−1 ) · (σ2 σ1 ) · (σ1 σ2 ), and thus R (x) = 8 = 2(x). But (Δ) − 1 = 2 only when N = 3. Theorem 17 shows that R gives a better approximation than G , and gives a theoretical motivation for the results described in [9]. Having both experimental [9] and theoretical evidence for the superiority of R over G , we concentrate henceforth on the former. 4.2. Quasi-geodesics in embedded Garside groups We need not stop here, and may consider, as in the case of BN , two distinct Garside structures of the same group, such that one of them embeds in the other. Let M1 , M2 be Garside monoids with fundamental elements Δ, δ, respectively, such that each atom of M1 is also an atom of M2 , and the group of fractions of M1 coincides with that of M2 . Then we may take a length in one Garside structure as an estimation for the length in the other. We will denote the used structure by a superscripted index. By Theorem 17, 2R (x)

≤

(2 (δ) − 1)2 (x) ≤ (2 (δ) − 1)1 (x);

1R (x)

≤

(1 (Δ) − 1)1 (x).

Thus, if 2 (δ) < 1 (Δ), 2R (x) has a smaller approximation factor at its upper bound. For the lower bound, let A2 be the set of atoms of M2 , and set α = max{1 (a) : a ∈ A2 }. Then 1 (x) ≤ α2 (x), and thus 1 (x) ≤ α2 (x) ≤ α2R (x).

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Theorem 21. In the above notation, 1 1 (x) ≤ 2R (x) ≤ (2 (δ) − 1)1 (x). α

The advantage of Theorem 21 is that the distortion factors are symmetrized around the used length function 2R (x). Our main application is the following. 4.3. The case of the braid group + as well as by Consider the braid group as generated by the Artin monoid BN + the BKL monoid BKLN (Examples 11–12), and let Δ and δ be their respective fundamental elements. Consider the minimal lengths 1 for the Artin structure, and 2 for the BKL structure of BN , respectively. 1 (Δ) = N (N − 1)/2, whereas 2 (δ) = N − 1. For each atom at,s of BKL+ N, 1 (at,s ) ≤ 2(t − s − 1) + 1 = 2(t − s) − 1. In particular, the maximum α of all these lengths satisﬁes

α ≤ 2N − 3. 2R ,

the length in BKL generators of the rational By Theorem 21, we have that normal form in the BKL structure of BN , is quite symmetrically close to the minimal Artin length: Corollary 22. For each x ∈ BN : 1 1 (x) ≤ 2R (x) ≤ (N − 2)1 (x). 2N − 3

For comparison, measuring the minimal Artin length by working solely with the Artin structure of BN , we only have (by Theorem 17): 1 (x) ≤ 1R (x) ≤ (1 (Δ) − 1)1 (x) =

N2 − N − 2 1 (x). 2

The gain may be viewed as follows: In the latter case, we have a constant (in N ) error factor from below, and quadratic error from above. In Corollary 22, both errors are linear, that is, the errors are symmetrized by dividing by O(N ) terms. Another matter, which we cannot prove at present, is that the lower bound in Corollary 22 seems to be a big underestimate in the generic case. It seems to us that in the generic case, the lower bound factor should not be much smaller than 1 (indeed, it may be greater than 1). In summary, we have theoretical evidence suggesting that estimating the minimal length in Artin generators by using rational BKL normal form should be better than the same estimation using rational Artin normal form. We now turn to experimental results concerning the random case.

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5. Experimental results 5.1. Initial experiments For the Artin presentation, it is shown in [9] that the rational Artin length is much better than greedy Artin length, at least with regards to solving random equations with diﬃcult parameters. Our initial experiments showed that this is also the case for the BKL presentation: The rational BKL length is better than greedy BKL length. In the initial phase of this project, we have compared various length functions induced by various alternative ways of measuring lengths of elements, and found out that only the rational BKL length outperforms the rational Artin length when the problem’s parameters are getting diﬃcult. The remainder of this report is therefore dedicated to the comparison of the these two leading candidates. 5.2. A detailed comparison We adopt the basic framework of [1, 10, 9]: The equations are in a ﬁnitely generated group G = a1 , . . . , ang ≤ Bns , where ns denotes the number of strings and ng denotes the number of generators of G. Each generator ai is a word in Bns obtained by multiplying wl (word length) independent uniformly random elements of {σ1 , . . . , σns−1 }±1 . In G, we build a sentence X of length sl (sentence length): X = a1 a2 · · · asl (For the while, we restrict sl ≤ ng.) Some of the ai -s may be equal, but we did not force that intentionally. We begin with a description of a test suitable for groups G which are close to being free. For each i ∈ {1, . . . , ng} and each ∈ {1, −1}, we give the generator ai the score (a− i X), sort the generators according to their scores (position 1 is for the shortest length), and reorder each block of identical scores by applying a random permutation. We then keep in a histogram the position of a1 . We do one such computation for each sample of G and X. While a1 a2 · · · asl is not the way a random sl sentence in G was deﬁned, this does not make the problem easier: We use each group G to produce only one such sentence. To partially compensate for the fact that G need not be free, we do the following. There could be several i ∈ {1, . . . , ng} such that X = ai a1 · · · ai−1 ai+1 · · · asl . Let COR denote the set of these ai , the correct ﬁrst generators. After sorting all generators as above, instead of looking for the position of a1 , we look at the lowest position an element of COR attained. Remark 23. A more precise, but infeasible, way to construct COR would be to ﬁnd all shortest presentations of X as a product of elements from {a1 , . . . , am }±1 , and let COR be the set of the ﬁrst generators in these presentations. For the parameters

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we have checked, we believe that this should not make a big diﬀerence. The results in Section 5.6 support this hypothesis. We have also checked one set of cases where sl > ng. In these cases we deﬁned X = ai1 ai2 · · · aisl , where ij = (j −1 mod ng)+1 for j = 1, . . . , sl, and made the obvious adjustments. In summary, for each set of parameters (ns, wl, ng, sl) mentioned below, and for being either the rational Artin or the rational BKL length, we have repeated the following at least 1, 000 times: Choose a1 , . . . , ang , compute X, compute COR, sort all generators ai according to the lengths (a− i X), ﬁnd the lowest position attained by an element of COR, and store this position number in the histogram. After dividing the numbers in the histogram by the numbers of samples made, we obtain the distribution of the best position of a correct generator. In light of the intended application described in the ﬁrst two sections, a natural measure to the eﬀectiveness of is the graph of the accumulated probability, showing for each x = 1, . . . , 2ng the probability that some correct generator attained a position ≤ x. The results of our experiments are divided into 4 sets such that in each set of experiments, only one parameter varies. This shows the eﬀect of that parameter on the diﬃculty of the problem. The varying parameter takes 3 possible values, so we have 3 pairs (since there are two length functions) of graphs. Each pair of graphs has its own line style, so to allow plotting all 6 graphs on the same ﬁgure. For all pairs, one of the graphs is always above or almost the same as the other. Fortunately, in all cases, it is the rational BKL length which is above the rational Artin length, so there is no need to supply this information in the ﬁgure. Finally, since the accumulated distributions all reach 1 for x = 2ng, the graphs are more interesting for the smaller values of x. We therefore plot only the ﬁrst 35 values of x. 5.3. When the sentence length varies Fix ns = 64, wl = 8, ng = 128. Figure 1 shows the accumulated probabilities for sl ∈ {32, 64, 128}. 5.4. When the word length varies For ns = sl = 64, ng = 128, and wl ∈ {8, 16, 32}, we obtain the graphs in Figure 2. The problem gets easier when wl increases, since this way G gets closer to a free group (where the length approach is optimal). The remarkable observation is that the harder the problem becomes (by making wl smaller), the greater the improvement of the rational BKL length over the rational Artin length becomes. 5.5. When the number of generators varies Now set ns = sl = 64, wl = 8, and let ng ∈ {32, 64, 128}. The graphs appear in Figure 3. Here too, the more diﬃcult the problem becomes (by increasing the number of generators), the greater the advantage of BKL over Artin is. Moreover,

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Y SL=32 SL=64 SL=128

1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10

X 0

5

10

15

20

25

30

Figure 1. When sl varies Y WL=8 WL=16 WL=32

1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10

X 0

5

10

15

20

Figure 2. When wl varies

25

30

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Y NG=32 NG=64 NG=128

1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10

X 0

5

10

15

20

25

30

Figure 3. When ng varies Y NS=16 NS=32 NS=64

1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10

X 0

5

10

15

20

Figure 4. When ns varies

25

30

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the graphs show that doubling ng has little inﬂuence on the performance of the rational BKL length, whereas it seriously degrades the performance of the rational Artin length. 5.6. When the number of strings varies Finally, set wl = 8, sl = 64, ng = 128, and let ns ∈ {16, 32, 64}. Here, the problem becomes easier when we increase ns (Figure 4). This is not in accordance with earlier results in [9, 10], and is perhaps due to the fact that we allow any correct generator, whereas in the earlier works we only counted a1 a success. Indeed, the more strings there are, the greater the chances are that words of length 8 commute. On the other hand, the graphs show that while the BKL approach beneﬁts a great deal when the number of strings is doubled, this is not quite so for the Artin approach. This means that the improvement in success rates due to commuting generators is not substantial.

6. Concluding remarks and proposed future research Memory-length algorithms give a powerful heuristic method to solve arbitrary equations in noncommutative groups, and consequently a variety of otherwise intractable problems. These algorithms rely on a good length function on the group in question. In the past, greedy Artin length was used as a length function on the braid group, and it was realized that rational Artin length gives better results. In this paper, we suggested to use rational BKL length to measure the minimal Artin length, and gave theoretical as well as experimental evidence for the advantage of the new function over rational Artin length, at least when randomization is modelled as in [1]. The main drawback in our estimations is that they give much larger lengths than the minimal length. Some interesting directions for possible improvements are: 1. As we have seen, the rational form can be computed from the greedy normal from by “removing” δ-s from the leading simple elements. We may be more greedy, and remove the available δ-s from the (leftmost) longest simple elements in the greedy normal form.4 This gives a new normal form in BN , which has shorter length in terms of atoms. The resulting length function may be yet better than the one proposed here. 2. For each x and each proposal for a length function of x, we can take the minimum of the lengths of several elements whose minimal length is not k smaller than that of x, including: x, x−1 , xδ for each k = 1, . . . , m − 1, m where m is the minimal with δ central. 3. Since we use left-oriented normal forms in our estimations, we can also try the corresponding right-oriented normal forms, and take the minimum. 4 This

was suggested to us by Uzi Vishne.

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4. We can iterate conjugation by δ and inverses (and other operations which are not increasing the minimal length) with shortening heuristics like Dehornoy handle-reduction. In [13] this was done only to a very limited extent. 5. In [13], Dehornoy handle-reduction was applied to the greedy normal form to obtain an estimation of the minimal length. We conjecture that applying Dehornoy handle-reduction to the rational normal form would give better estimations. Acknowledgement We thank Joan Birman and Dima Ruinskiy for their comments on earlier versions of the paper. We also thank Patrick Dehornoy and Sang Jin Lee for informative discussions concerning our notation, and the referees for their useful comments. A special thanks is owed to Arkadius Kalka for useful discussions and suggestions.

References [1] I. Anshel, M. Anshel and D. Goldfeld, An algebraic method for public-key cryptography, Math. Res. Lett. 6 (1999), 287–291. [2] M. Berger, Minimum crossing numbers for 3-braids, Journal of Physics A: Mathematical and General 27 (1994), 6205–6213. [3] J. Birman, K.H. Ko, J.S. Lee, A new approach to the word and conjugacy problems in the braid groups, Advances in Mathematics 139 (1998), 322–353. [4] R. Charney, Geodesic automation and growth functions for Artin groups of ﬁnite type, Mathematische Annalen 301 (1995), 307–324. ´ [5] P. Dehornoy, Groupes de Garside, Annales Scientiﬁques de l’Ecole Normale Sup´erieure 35 (2002), 267–306. [6] P. Dehornoy and L. Paris, Gaussian groups and Garside groups, two generalisations of Artin groups, Proceedings of the London Mathematical Society 79 (1999), 569– 604. [7] D. Epstein, J. Cannon, D. Holt, S. Levy, M. Paterson, and W. Thurston, Word Processing in Groups, Jones and Bartlett Publishers, Boston: 1992. [8] D. Garber, Braid group cryptography, www.ims.nus.edu.sg/Programs/braids/files/david.pdf [9] D. Garber, S. Kaplan, M. Teicher, B. Tsaban, and U. Vishne, Length-based conjugacy search in the Braid group, Contemporary Mathematics 418 (2006), 75–87. [10] D. Garber, S. Kaplan, M. Teicher, B. Tsaban, and U. Vishne, Probabilistic solutions of equations in the braid group, Advances in Applied Mathematics 35 (2005), 323– 334. [11] J. Hughes and A. Tannenbaum, Length-based attacks for certain group based encryption rewriting systems, Workshop SECI02 S´ecurit´e de la Communication sur Internet, September 2002. [12] K.H. Ko, S.J. Lee, J.H. Cheon, J.W. Han, S.J. Kang and C.S. Park, New Publickey Cryptosystem using Braid Groups, CRYPTO 2000, Lecture Notes in Computer Science 1880 (2000), 166–183.

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[13] A. Myasnikov, V. Shpilrain, and A. Ushakov, A practical attack on some braid group based cryptographic protocols, in: CRYPTO 2005, Lecture Notes in Computer Science 3621 (2005), 86–96. [14] A. Myasnikov, V. Shpilrain, and A. Ushakov, Group-based cryptography, Advanced Courses in Mathematics – CRM Barcelona, Birkh¨auser, 2008. [15] M. Paterson and A. Razborov, The set of minimal braids is co-NP-complete, Journal of Algorithms 12 (1991), 393–408. [16] D. Ruinskiy, A. Shamir, and B. Tsaban, Length-based cryptanalysis: The case of Thompson’s Group, Journal of Mathematical Cryptology 1 (2007), 359–372. Martin Hock Department of Computer Science University of Wisconsin Madison, WI 53706, USA e-mail: [email protected] Boaz Tsaban Department of Mathematics Bar-Ilan University Ramat-Gan 52900, Israel and Department of Mathematics Weizmann Institute of Science Rehovot 76100, Israel e-mail: [email protected] URL: http://www.cs.biu.ac.il/~tsaban

Combinatorial and Geometric Group Theory Trends in Mathematics, 171–202 c 2010 Springer Basel AG

An Application of Word Combinatorics to Decision Problems in Group Theory Arye Juh´asz Abstract. In this work we develop a graph theoretical test on graphs corresponding to subgroups of one-relator groups with small cancellation condition which, if successful, implies that the subgroup under consideration has solvable membership problem with a simple solution. The proof of the solvability of the membership problem relies on word combinatorics in an essential way. Mathematics Subject Classiﬁcation (2000). Primary 20E06; Secondary 20F05, 20F06. Keywords. One-relator groups, small cancellation conditions, membership problem, Whithead graphs.

Introduction In his seminal paper [M], Wilhelm Magnus solved the word problem for one relator groups. Brieﬂy, he showed that one-relator groups which are generated by at least two elements have the structure of HNN-extensions with Magnus subgroups as special subgroups (Magnus subgroups are groups generated by proper subsets of the given generators of the group). Then he solved the membership problem for Magnus subgroups. From this one easily gets a solution of the word problem. Recall that the membership problem (M.P.) asks for an algorithm to decide whether a given element of the group belongs to a given subgroup. In spite of the long period passed from the occurrence of [M], very few new results are in the literature about the membership problem in one-relator groups. This problem is not known to be solvable even in hyperbolic one-relator groups. In the present work we exhibit a class of subgroups in one-relator groups with small cancellation, which are not Magnus subgroups and for which the membership problem is solvable. Whether a given subgroup belongs to this class is determined by a graph attached to the subgroup. Thus, let G be a one-relator group given by ﬁnite presentation P = X|R, where R is the symmetric closure of a single cyclically reduced word R. Let F (X) be the free group freely generated by X. Let

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H be a subgroup of F (X) and let H be its image in G. Recall that the Whitehead graph of a word W in F (X) is, by deﬁnition, the graph on 2|X| vertices which are labeled by elements of X ∪ X −1 in which two vertices, say v1 , with label a1 and v2 with label a2 (a1 , a2 ∈ X ∪ X −1 ), are connected by an edge if a−1 1 a2 is a subword of W or of W −1 [L-S]. We shall denote by Wh(W ) the graph obtained from the Whitehead graph by identifying edges with the same endpoints. If T is a set of words, we denote by Wh(T ) the graph with vertices X ∪ X −1 and two vertices u and v are connected by an edge if and only if u−1 v or v −1 u is a subword of a word of T . Our Main Theorem loosely says that if Wh(H) does not contain a large portion of Wh(R), then H has solvable membership problem. Denote by E(W ) the set of edges of Wh(W ) and similarly let E(T ) the set of ˆ the cyclic word corresponding to R. edges of Wh(T ). Denote by R Main Theorem. Let P = X| R be a one-relator presentation of a one-relator group G with |R| ≥ 5 which satisﬁes the small cancellation conditions C (1/5) and T (4). Let F = F (X) and let θ : F → G be the canonical homomorphism. Let Y ⊆ F be a ﬁnite subset of F , let H be the subgroup of F generated by Y and let and let K = E(H). If H = θ(H). Let Z = E(R) |Z ∩ K| < |Z| − 3

(∗)

then H has solvable Membership Problem in G. Two remarks are in order here. First, with the cost of more work on the side of small cancellation theory, we can replace the metric condition C (1/5) by the combinatorial condition C(6). Next, when checking condition (∗) we do not have to check each element of H when forming K, because there is a standard way by folding edges with a common endpoint and the same label in the graph which corresponds to H (see [L-S], p. 118) to construct a graph from which K can be easily read oﬀ. Example 0.1. Let F = a, b, c, d| − , R = P 2 U1 U2 , where P = ab−2 a2 c3 d−1 acb−1 d−2 a, U 1 = d−1 a1 4d−1 b−1 , U 2 = c2 b−1 d−1 cb−1 . Let H = U1 , U2 . Then H := θ(H) has solvable membership problem, by the Main of the cyclic word Theorem. To see this, consider ﬁrst the Whitehead graph W h(R) −1 ¯ R. It is depicted in Fig. 1, where a ¯ denotes a and similarly for ¯b, c¯ and d. 8 · 4 is a 4-regular graph with = 16 edges. Next, we consider the Thus, Wh(R) 2 Whitehead graphs of the elements of H. Since U1 and U2 have length greater than ε two and since the only cancellation in the products Uiεi Uj j , where εi , εj ∈ {1, −1}, i, j ∈ {1, 2} and i = j ⇒ εi = εj are in U1 U2−1 , in which case only one letter is cancelled out (b), hence the Whitehead graph of every element in H is contained in ε the union of the Whitehead graphs of Ui and Uiεi Uj j , εi , εj ∈ {1, −1}, i, j ∈ {1, 2}. = 16 These are marked in Fig. 1. by a tilde. Thus, we see that K ≤ 8. But |Wh(R)| and 8 < 16 − 3 = 13. Consequently, if R satisﬁes C(6)& T (4) then H has solvable

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173

Figure 1

membership problem in G, by the Main Theorem (see remark above). But since contains no closed curves of length less than or equal three, it follows that Wh(R) R satisﬁes T (4) and it is easy to check that it satisﬁes the condition C(6) as well. Thus, H has solvable membership problem in G, by the Main Theorem and the ﬁrst remark following the theorem. We ﬁnish this introduction with a few remarks on the method of proof. This work extends the methods and results of [J2], which we recall in Section 1 and the beginning of Section 2. We use small cancellation theory. A central ingredient of the theory is Greendlinger’s Lemma, which guaranties the existence of at least two Greendlinger regions in every van Kampen diagram M which has at least two regions. (For deﬁnitions of van Kampen diagrams and regions see 1.1.) These are regions with the property that their boundary has a large common portion with the boundary of M . Since the label of the boundary ∂M of M is a consequence of R and for every consequence there is such a diagram, this means in algebraic terms that every consequence of R contains a large portion of one of the deﬁning relations. (See [L-S, Ch. V].) If the small cancellation condition is strong enough, this result alone is enough to solve the word problem. However, the solution of our problem requires a more precise information than just a large portion of the deﬁning relator. In the present work we extend Greendlinger’s Lemma in order to get the more precise information by describing the relationship between Whitehead graphs of deﬁning relators and their consequences, using massive word combinatorics. The appearance of word combinatorics in the context of small cancellation theory is quite natural: for example, there are standard results, like Lyndon-Sch¨ utzenberger’s Lemma for periodic words, which guarantees that long subwords which occur more than once in the word, occur in a very special conﬁgurations, which can be avoided by an appropriate small cancellation theory.

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Yet, the application of word combinatorics in the present work is of a diﬀerent nature: we apply word combinatorics in order to improve Greendlinger’s Lemma for the one-relator case. We exemplify this by the proof of Magnus’ Freiheitssatz for one-relator free products with several components; G = G1 ∗ · · · ∗ Gm | R, m ≥ 3 as given in [J2]. (This problem is similar to ours, but much simpler.) We have to show that every consequence of R contains a letter from each Gi , i = 1, . . . , m. Let C be a consequence of R and let M be a van Kampen diagram with C as a boundary label. (See 1.1.) It follows by standard small cancellation theory that under the condition C (1/5)&T (4) a diagram M has a Greendlinger region D which either has only one neighbouring region E in M or it has two neighbours Er and E which have common boundary paths with D, having labels P1 and P2 , respectively. (See Fig. 8(a) and Fig. 9, respectively.) Suppose for simplicity the ﬁrst and let P be the label of the common boundary path ∂E ∩ ∂D of E and D and let Q be the label of the common boundary path of D and M . Thus R1 := QP is a boundary label of D, hence a cyclic conjugate of Rε , ε ∈ {1, −1}. We claim that every letter in P necessarily occurs in Q. Now P is a common label of ∂D and ∂E. Therefore, P occurs as a subword of R1 and also as a subword of a boundary label R2 of E. But since R is the only deﬁning relator of G, R1 and R2 are cyclic conjugates of R±1 . Therefore, in addition to the above-mentioned 1 , which we 1 , P ε , ε ∈ {1, −1}, has another occurrence in R occurrence of P in R denote by P ; it comes from the occurrence of P as a subword of R2 which is a cyclic conjugate of R1 . (These occurrences are diﬀerent because we assume that our diagrams are reduced.) Now, since R1 = QP and P is a subword of the cyclic 1 , either P is a subword of Q in which case we are done, or else P overlaps word R non-trivially with P . See Fig. 8 (b). In this case ε = 1 and we have the following word equations: P = XY , P = AX and Q = Y Q1 . In particular, XY = AX(= P ) hence by the well-known (and easy) result from word combinatorics we get that P = (KL)α K, α ≥ 1 and Y = KL, for certain subwords K and L of P . But then Q = Y Q1 = KLQ1 and since all the diﬀerent letters occurring in P already occur in KL, we get that all the letters in P occur in Q, as required. Observe that we used here word combinatorics in order to shift the letters of P which occur inside the diagram into Q, which occurs on the boundary of the diagram. In the case when D has two neighbours Er and E it is not always true that Q contains all the letters of P1 and P2 . In fact we prove that Q QQr contains every letter of R, where Qr and Q are tails and head of the labels of ∂Er ∩ ∂M and ∂E ∩ ∂M , respectively.

(∗)

This requires the development of a rather complicated combinatorial machinery on words which we develop in Section 2 of the present paper. See Appendix for full details. We describe it very brieﬂy. First, observe that in the above word equations we were interested not so much in their solution but rather the values of the function Supp : F → L, on the solutions where L is the set of all the subsets of {1, . . . , m} and for a word W in F, Supp(W ) is the set of the indices i for

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which Gi contributes a letter in W . We rely very heavily on this observation when proving (∗). In the present work we deal with Whitehead graphs hence we have to introduce a new function σ : F → K, where K is the set of all the subsets of the set of words in F with length two. (This corresponds to the edges of the corresponding Whitehead graph.) Finally, we point out that the above-mentioned improved version of Greendlinger’s Lemma holds true under the small cancellation condition C(6), with a few known exceptions, however, the corresponding word combinatorics is incomparably much more complicated. The paper is organised as follows. In Section 1 we introduce the necessary preliminaries from small cancellation theory. In Section 2 we develop the word combinatorics needed. In Section 3 we prove the Main Theorem.

1. Preliminaries on small cancellation theory 1.1. Diagrams One of our main tools is van Kampen diagrams. These are planar complexes introduced by van Kampen in 1933, which give precise description of the way cancellation carried out when forming the right-hand side of (1.2) below. For basic results on diagrams see [L-S, Ch. V]. We recall here some of the basic deﬁnitions and results from [L-S, pp. 237–239 and pp. 274–276], for convenience. Let E2 denote the Euclidean plane. If S ⊆ E2 then ∂S will denote the boundary of ¯ A vertex is a point of E2 . An S, the topological closure of S will be denoted by S. 2 edge is a bounded subset of E homeomorphic to the open unit interval. A region is a bounded set homeomorphic to the open unit disc. A map is a ﬁnite collection of vertices, edges and regions which are pairwise disjoint and satisfy: (i) If e is an edge of M , there are vertices u and v (not necessarily distinct) in M such that e¯ = e ∪ {u} ∪ {v}. (ii) The boundary, ∂D, of each region D of M is connected and there is a set of edges e1 , . . . , en such that ∂D = e¯1 ∪ · · · ∪ e¯n . A diagram over a group F is an oriented map M and a function Φ assigning to each oriented edge e of M as a label an element Φ(e) of F such that if e is an oriented edge of M and e−1 the oppositely oriented edge, then Φ(e−1 ) = Φ(e)−1 , and if μ = e1 v1 e2 v2 · · · ek is a path in M then Φ(μ) = Φ(e1 )Φ(e2 ) · · · Φ(ek ). We denote by ΦM the labelling function of M over F . If M is ﬁxed we shall write Φ for ΦM . If M is a diagram then its underlying map is the map obtained from M by ignoring the labels. For a closed path μ ¯ in M denote by h(μ) its initial vertex and by t(μ) its terminal vertex. Now, the connection between diagrams and consequences of relators is given by the following two theorems.

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Theorem 1.1. [L-S, p. 237] If F is a free group and C1 , . . . , Cn , n ≥ 0 is a sequence of non-trivial elements of F (the conjugates of relators), there is a diagram M (C1 , . . . , Cn ) which satisﬁes each of the following: (i) if e is an edge of M then Φ(e) = 1; (ii) M is connected and simply connected, with a distinguished vertex 0 on ∂M ; (iii) there is boundary cycle e1 · · · · · et of M beginning at 0 such that the product Φ(e1 ) · · · · · Φ(et ) is reduced as written and in the free group we have Φ(e1 ) · · · · · Φ(et ) = C1 · · · · · Cn

(1.1)

Diagrams satisfying conditions (i), (ii) and (iii), are called van Kampen diagrams. The next theorem provides a converse of Theorem 1.1. Theorem 1.2. (Normal Subgroup Theorem) [L-S, p. 239] Let M be a connected, simply connected diagram with regions D1 , . . . , Dm . Let α be a boundary cycle of M beginning at a vertex v0 ∈ ∂M and let W = Φ(α). Then there exist labels Ri of Di and elements fi of F for 1 ≤ i ≤ m such that −1 W = f1−1 R1 f1 · · · · · fm R m fm

(1.2)

Example 1.3. (See [Jo, p. 214].) Consider the (by now familiar) group " ! G = x, y | x2 yxy 3 , y 2 xyx3 . The non-obvious fact that x7 = e in G is embodied in the following diagram, where W = x7 .

Figure 2

Deﬁnitions 1.4. Let M be a diagram over F . (a) Two regions D1 and D2 in M are neighbours if ∂D1 ∩ ∂D2 = ∅. They are proper neighbours if ∂D1 ∩ ∂D2 contains a non-empty edge. For example in Fig. 2. D1 and D2 are proper neighbours, while D2 and D3 are neighbours, but not proper neighbours.

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(b) A region D is a boundary region if ∂D ∩ ∂M = ∅. A region D is a proper boundary region if ∂D ∩∂M contains a non-empty edge. A region of M which is not a boundary region is an inner region. For example in Fig. 2. D2 is an inner region, D3 is a proper boundary region, while D1 is a boundary region, but not proper. (c) A neighbour E of a boundary region D is an inner-neighbour if E is an inner region. Deﬁnition 1.5. Let M be a connected, simply connected map. M is a simple onelayer map, if the dual map M ∗ , obtained from M by putting in each region D a vertex D∗ and connecting two vertices D1∗ and D2∗ by an edge if D1 and D2 are proper neighbours, is a straight line. (See Fig. 3(b).) In particular, M has connected interior, every region is a boundary region, each region has at most two proper neighbours and if M contains more than one region then M contains exactly two regions (D1 and Dr on Fig. 3(b)) which have exactly one neighbour each. M is a one-layer map if it is composed from simple one-layer maps and paths in the way shown on Fig. 3(a).

(a)

(b) Figure 3 If M is a simple one layer map consisting of regions D1 , . . . , Dt which occur in this order from left to right or from right to left then we shall denote this by

D1 , . . . , Dt . 1.2. Diagrams with small cancellation conditions We recall the following result from [L-S, p. 274]. Lemma 1.6. Let P = X|R be a presentation and let M be a van Kampen diagram, the boundary label of which is a consequence of R.

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(a) If P satisﬁes the condition C(p) for p ≥ 3, then every inner region of M has at least p proper neighbours. (b) If P satisﬁes the condition T (q) for q ≥ 3, then every inner vertex of M has valency at least q. By analogy, we shall say that a map satisﬁes the condition C(p) if every inner region has at least p proper neighbours and a map satisﬁes the condition T (q) if every inner vertex has valency at least q. Thus if M is the underlying map of a diagram, the boundary label of which is a consequence of R and P satisﬁes the condition C(p) and T (q) then M also satisﬁes these conditions. We recall the main structure theorem for C (1/5) and T (4) maps from [J1], where it is proved in a more general setting. (Observe that condition C (1/5)& T (4) implies condition W (6) in [J1].)

(a)

(b) Figure 4

Theorem 1.7 (Layer Decomposition). [J1] (See Fig. 4.) Let M be a simply connected map (diagram) with connected interior and let D0 be a region of M . Assume that M satisﬁes the condition C(6)& T (4). Deﬁne St0 (D0 ) = D0 and let Sti (D0 ) = Sti−1 (D) ∪ Li (D0 ) for i ≥ 1, where L0 (D0 ) = {D0 }, ! " Li (D0 ) = D in M \Sti−1 (D0 ) ∂D ∩ ∂Sti−1 (D0 ) = ∅ . Let p be the smallest number such that Stp (D0 ) = M and assume that p > 0 (i.e., M contains more than one region). Then each of the following holds: (a) Every regular submap of Sti+1 (D0 ) containing Sti (D0 ) for 0 ≤ i ≤ p is simply connected. (A submap is regular if every edge is on the boundary of a region.) (b) Every connected and simply connected submap of Li (D0 ) is a one-layer map. When D0 is ﬁxed, we shall abbreviate Li (D0 ) by Li and call Λ(D0 ) = (L0 , . . . , Lp ) a layer decomposition of M . We call D0 the center of the layer decomposition.

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(c) For a region D ∈ Li , i ≥ 1 denote by A(D) the set of regions E in Li−1 , which have a non-trivial common edge with D, denote by B(D) the set of regions S in Li with ∂S ∩ ∂D = ∅. Also, let a(D) = |A(D)|, b(D) = |B(D)|. Then a(D) ≤ 1 and b(D) ≤ 2. In other words, D has at most two neighbours in Li and at most one neighbour in Li−1 . (d) If v ∈ ∂Sti (D0 ) then v has valency at most three in Sti (D0 ). (e) For regions D, E in M with ∂D ∩ ∂E = ∅ we have that ∂D ∩ ∂E is connected. Remark 1.8. Let M be a connected simply connected map (diagram) with connected interior and let D be a region in M . Let Λ(D) be a layer decomposition of M with center D. Suppose that D is a boundary region of M with a non-empty edge on ∂M . (See Fig. 5.) Then it follows from the above theorem that L1 (D) is not annular, hence simply connected, though not necessarily with connected interior. (See Fig. 5(a), where the interior of L1 is simply connected and connected and see Fig. 5(b), where the interior of L1 is not connected.) But then due to the simply connectedness of M , Li is simply connected for every i.

(a)

(b) Figure 5

Remark 1.9. We could start the construction of the layers not with a region D0 , but with a vertex v and deﬁning St0 (v) = {v}, St1 (v) is the submap consisting of {D v ∈ ∂D} and for k ≥ 2 deﬁne Stk (v) like in Theorem 1.7. Moreover, we can deﬁne a layer decomposition relative to a path in M , under certain conditions. Thus, a center may be either an arbitrary vertex, or an arbitrary region, or a path of a special type which we call a transversal. (See Deﬁnition 1.19.) Remark 1.10. The notion of one-layer map is independent of any given layer decomposition. For example the map on Fig. 3(b) is a one-layer map, by deﬁnition. However, it also has a layer decomposition Λ with center D1 for which L1 consists of D2 and D3 . Another typical example is Mk on Fig. 3(a), which is a one-layer map. In its layer decomposition with center D0 every layer consists of a single region, hence certainly each Li is a one-layer map, however these layers Li are “perpendicular” to the natural one-layer structure of Mk .

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In the next deﬁnition we introduce special subdiagrams and regions, the boundaries of which share a large portion with the boundary of M . Deﬁnition 1.11. Let Λ(D0 ) be a layer decomposition of M with center D0 on the boundary of M such that if D0 is a region then it has a non-empty edge on ∂M . If Λ(D0 ) = (L0 , L1 , . . . , Lp ) then the closure of every connected component P of the interior of Lp is a peak. We say that P is a peak relative to D0 . (See Fig. 4(a), where p = 2 and L2 (D0 ) is a peak and See Fig. 4(b), where Lp (D0 ) is a peak.) By Theorem 1.7 every layer structure has a peak. Related to peaks is the following notion. Deﬁnition 1.12. A boundary region D of M is a k-corner region for k = 1, 2 if each of the following holds: 1) ∂D ∩ ∂M is connected and 2) D has k inner proper neighbours. Deﬁnition 1.13. Let μ be a boundary path of M , let P be a peak of M relative to D0 and let D be a k-corner region with k ≥ 2. Say that μ contains P if ∂P ∩ ∂M ⊆ μ. Similarly, μ contains D if ∂D ∩ ∂M ⊆ μ. Lemma 1.14. Let D be a k-corner region in a diagram M with k ≤ 2, which satisﬁes the condition C (1/5). Let θ = ∂D ∩ ∂M and let η be the complement of θ on ∂D. (Thus, vθuη is a boundary cycle of D with v and u endpoints of θ and η.) Then |Φ(θ)| > |Φ(η)|. Proof. Since k ≤ 2, the region D has at most two proper neighbours, hence Φ(η) 1 1 is the product of at most two pieces. Therefore |Φ(η)| < 2 · |∂D| < |∂D|. Since 5 2 1 1 |Φ(θ)| = |R| − |Φ(η)|, |Φ(θ)| > |R| − |R| = |R| > |Φ(η)|, i.e., |Φ(θ)| > |Φ(η)|, 2 2 as required. The lemma is proved. Lemma 1.15. Let M be a map with a layer decomposition Λ(D0 ) and let P be a peak relative to D0 . Then P contains a k-corner region for some k ≤ 2. Proof. Let P = D1 , . . . , Dk , k ≥ 1. If k ≥ 2 then its extremal regions D1 and Dk are 2-corner regions because a(D1 ) ≤ 1 and b(D1 ) ≤ 1, due to being extremal and clearly ∂D1 ∩ ∂M is connected. Also, if a(Dk−1 ) = 0 then Dk−1 in Fig. 6(a) is a

(a)

(b) Figure 6

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2-corner region. If P is a peak consisting of a single region E, then E is a 1-corner region due to Theorem 1.7. The lemma is proved. The following lemma is an immediate consequence of Lemmas 1.14 and 1.15, hence we omit its proof. Lemma 1.16. Let N be the normal closure of R in F . Let M be a diagram for a consequence of R, which satisﬁes the condition C (1/5) and let μ be a boundary path of M . Suppose that μ contains a peak relative to D0 . Then Φ(μ) is not a shortest representative of the elements of Φ(μ)N . Deﬁnition 1.17. Let M be a connected, simply connected diagram with no vertices of valency one. Let μ be a boundary path of M and let v ∈ μ be a vertex. Say that v is a double point of μ if v is met more than once when traversing along μ. (Actually it may be a multiple point, but from our point of view all we need is that it is not regular.) Remark 1.18. It is well known (see [L-S, Ch. V]) that if μ has a double point then μ encloses one or more connected components of the interior of M . Hence, if μ contains a double point v then μ contains a peak relative to v, due to Theorem 1.7 and Remark 1.9. An immediate consequence of the last remark and Lemma 1.16 is the following: If Φ(μ) is a shortest representative of Φ(μ)N in G then μ contains no double points.

(∗)

We introduce now one of our main tools concerning diagrams, the transversals. Let Λ(D) be a layer decomposition of M and let ωi = ∂Li ∩ ∂Li+1 for i = 1, . . . , p − 1, see Fig. 7(a) showing ωi and ωi+1 . Then due to Theorem 1.7 ωi has the property that ∂E ∩ ωi is either a vertex or an edge, for every region E of Li+1 . (Observe that this is not true for regions in Li . If K is a region of Li then ∂K ∩ ωi may contain more than one edge.) It turns out that this property of ωi is responsible for the existence of peaks in M . We develop this idea below in order to produce peaks on speciﬁc places along ∂M . 1.3. Transversals in diagrams with Small Cancellation Conditions Let Λ(D) be a layer decomposition of M , where D may be a single vertex. Any vertex in M with valency at least three not in the last layer is on the common boundary ωi of Li (D) and Li+1 (D) for some i. Suppose that Li+1 (D) contains a vertex w1 ∈ ωi which has valency at least four in Li+1 (D). (See Fig. 7(a).) For example, if Li+1 (D) contains at least three regions, then, due to the C (1/5) & T (4) condition, it follows from Theorem 1.7(c) that Li+1 (D) contains such a vertex w1 . Let u be the initial vertex of ωi and let uθ1 w1 be the subpath of ωi which starts at u and terminates at w1 . Then, as observed above, due to Theorem 1.7(c) θ1 satisﬁes condition (∗∗) below, with μ = θ1

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(a)

(b) Figure 7

when traversing along μ, for every region E to the left of μ with ∂E ∩ μ nontrivial, (i.e., contains an edge) we have that ∂E ∩ μ is connected and ∂E ∩ μ is an edge (i.e., does not contain a vertex with valency at least three in M ).

(∗∗)

Since w1 has valency at least four in M , hence Li+1 (D) contains a region E with ∂E ∩ ωi = {w1 }. Let θ2 = ∂E ∩ ∂F , where F is the right-hand side neighbour of E in Li+1 (D). Then uθ1 w1 θ2 w2 satisﬁes condition (∗∗) above, where w2 is the endpoint of θ2 , diﬀerent from w1 . Assume that w2 is not a boundary vertex of M . Then w2 ∈ ωi+1 and there are left most adjacent regions E1 and F1 in Li+2 (D) which contain w2 on their boundary. Deﬁne θ3 = ∂E1 ∩ ∂F1 and keep deﬁning θi and wi for i ≥ 3 until reaching ∂M . Let θ = uθ1 w1 θ2 w2 . . . θe v where v is a boundary vertex of M and wi are inner vertices of M for 1 ≤ i ≤ e − 1. Then v ∈ ∂M and each initial segment of θ satisﬁes condition (∗∗). This leads us to the following deﬁnition. Deﬁnition 1.19. Let notation be as above. Let θ be a simple path in M with initial and terminal vertices u and v, respectively. Suppose that u, v ∈ ∂M and θ ∩∂M = ∅ (i.e., the open path θ does not intersect ∂M ). Call θ a left transversal if it satisﬁes condition (∗∗) above. If μ is a left transversal, then μ deﬁnes a submap Mμ of M by its boundary uμvζ −1 where ζ is the boundary path of M which starts at u, terminates at v and is to the left of μ. (See Fig. 7(a).) The proof of Theorem 1.7 (see [J1]) can be easily adopted to prove the next proposition. We omit its proof here. Proposition 1.20. Let μ be a left transversal in M . Then Mμ has a layer structure relative to μ. In particular, Mμ has a peak relative to μ. More precisely, deﬁne

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L0 (μ) = μ. Let L1 (μ) be the submap of Mμ consisting of those regions of Mμ , the boundary of which intersects μ. Then L1 (μ) is connected and simply connected, and μ is a boundary path of L1 (μ). Let μ1 be its complement on ∂L1 (μ). Then μ1 = α1 μ1 α−1 2 , where α1 and α2 are boundary paths of M and μ1 is an inner path of M . Deﬁne L2 (μ) = L1 (μ1 ) and repeat this process until the last layer Lk (μ) is reached. (See Fig. 7(b).) Then L0 (μ), . . . , Lk (μ) is a layer structure of Mμ relative to μ. Clearly, by the same method we may construct right transversals. Hence, if Li+1 (D) contains at least two vertices with valency at least four in Li+1 (D) or one vertex with valency at least ﬁve in Li (D) and u, u are endpoints of ωi , say u to the left of u , then we may construct a left transversal μ with initial vertex u and a right transversal ν with initial vertex u . (See Fig. 7(a).) It follows by an easy induction on the number of layers that Mμ ∩ Mν = ∅. Now, due to the C (1/5) & T (4) condition and Theorem 1.7(c), if Li+1 (D) contains at least four regions which have a non-empty common edge with ωi , then Li+1 (D) contains at least two vertices with valency at least four. We summarize this in the following proposition. Proposition 1.21. Let M be a diagram, let D be either a boundary region of M such that ∂D ∩ ∂M contains a non-empty edge or a boundary vertex, and let Λ(D) be a layer decomposition of M with center D. Suppose that Li+1 (D), (i ≥ 1) contains at least four regions which have a non-empty edge on ωi := Li (D) ∩ Li+1 (D) or a vertex with valency at least ﬁve in Li+1 (D). Then (a) M contains a right transversal ν and a left transversal μ with Mμ ∩ Mν = ∅ such that ν starts at t(ωi ) and μ starts at h(ωi ). (See Fig. 7(a).) (b) Let ζμ be the boundary path of M with initial vertex h(μ) and terminal vertex t(μ) which is to the left of Mμ and similarly, let ζν be the boundary path of M which starts at h(ν) and ends at t(ν) and is to the right of Mν . Then each of ζμ and ζν contains a peak. 1.4. The Main Theorem for almost σ-complete presentations We are now in a position to prove the Main Theorem under some additional assumptions, deﬁned below. The main result of Section 3, which is mostly word combinatorics is that this assumptions are satisﬁed under the conditions of the theorem. Deﬁnitions 1.22. Let R be a cyclically reduced word, let R be the symmetric closure of R and let P = X|R be a one-relator presentation. Let M be an R-diagram with layer decomposition Λ and let P be a peak relative to Λ, with α = ∂P ∩ ∂M . (a) For every word W denote by σ(W ) the set of all the subwords of W of length two. with (b) P is almost σ-complete if σ Φ(α) ⊇ Γ for some subset Γ of σ(R), − 3. |Γ| ≥ |σ(R)|

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(c) M is almost σ-complete if every peak relative to every layer decomposition of M is almost σ-complete. (d) P is almost σ-complete if every R-diagram M is almost σ-complete. The next lemma shows the signiﬁcance of Deﬁnition 1.22. Lemma 1.23. Let notation be as in Deﬁnition 1.22. Let H be a subgroup of F (X) which satisﬁes condition (∗) of the Main Theorem. Let M be a van Kampen diagram for a consequence of R and let ν be a boundary path of M with V := Φ(ν) ∈ H. If M is almost σ-complete then ν contains no peaks. In particular, ν contains no double points. Proof. If ν contains no peaks then it cannot contains double points, due to Theorem 1.7. (See Remark 1.18.) Hence it is enough to show that ν contains no peaks. It follows from the assumption of the theorem that then |Γ| < |σ(R)| −3 if σ(V ) contains a non-empty subset Γ of σ(R) for every reduced word V in H.

(∗)

If P is a peak relative to a layer decomposition Λ and α = ∂P ∩ ∂M , then by the additional assumption that P is almost σ-complete it follows that for some non we have |Γ| ≥ |σ(R)| − 3. empty subset Γ of σ Φ(α) which is contained in σ(R), of length greater than |R| − 2 |R| = 3 |R| ≥ 3, Since Φ(α) contains a subword of R 5 5 due to the C (1/5) and T (4) conditions and Lemma 1.14, hence σ Φ(α) contains But since α is a subpath of ν by assumption, this a non-empty subset Γ of σ(R). contradicts the fact that M is almost σ-complete, proving the lemma. In this subsection we prove the Main Theorem for almost σ-complete onerelator presentations with small cancellation condition C (1/5)& T (4). Theorem 1.24. Let notation and assumptions be as in the Main Theorem. If P is almost σ-complete then the result of the Main Theorem holds true. Proof. We keep the notation of the Introduction and the Main Theorem. Also we assume P is almost σ-complete. We start with the remark that G has solvable word problem by [L-S, p.262] Our proof is based on the following proposition. Proposition 1.25. Let M be a connected, simply connected R-diagram. Let μ be a boundary path of M and let ν be its complement on ∂M . (Thus, uμvν −1 is a boundary cycle of M , where u and v are vertices.) Let Φ(μ) = U and let Φ(ν) = V . Let U = u1 · · · up and let R = r1 · · · rq , reduced as written, ui , ri ∈ X ∪ X −1 . Let ±1 n S = u±1 i , rj , 1 ≤ i ≤ p, 1 ≤ j ≤ q ∪{1} and for every natural number n let S be the set of all the products of n elements from S. Suppose that U is a shortest representative of θ(U ) and V ∈ H. Then each of the following holds: (a) If η is a simple boundary path of M then Φ(η) ∈ S |η| . (b) |V | ≤ 9|U ||R| and V is a word on S, eﬀectively computable from U and R.

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Proof. Since U is a shortest representative of θ(U ), μ contains no double points, by (∗) in Remark 1.18. Also, by Lemma 1.23, ν contains no peaks. Consequently, M has the form like the diagram in Fig. 3(a). (a) If M1 , . . . , Mk are the connected components of Int(M ) then the subpaths ζi k for i = 0, . . . , k, k + 1, which are the connected components of M \ Mi , i=1

are subpaths of μ and ν. Hence, if e is an edge of M then one of the following holds for e: (i) e ⊆ ζi for some i, i = 0, . . . , k + 1; (ii) e is an edge of Mi , for some i, i = 1, . . . , k; We propose to show that in each of these cases Φ(e) ∈ S. This will clearly for some j, 1 ≤ j ≤ p, hence prove part (a). Now, in case (i) Φ(e) = u±1 j Φ(e) ∈ S. In case (ii) Φ(e) = rj±1 for some j, 1 ≤ j ≤ q, hence in particular Φ(e) ∈ S, as required. (b) Let Λ be the layer decomposition of M with center u. Assume Λ has s layers. For i = 1, . . . , s − 1 let ωi = ∂Li ∩ ∂Li+1 and for i = 1, . . . , s let μi = ∂Li ∩ μ and νi = ∂Li ∩ ν. Then μ = μ1 · · · μs , ν = ν1 · · · νs and by the deﬁnition of layers we have (1.1) 1 ≤ |μi | and 1 ≤ |νi |. Moreover, due to (1.1) s ≤ |μ| and s ≤ |ν|.

(1.2)

Suppose M has a layer Li+1 which contains at least four regions with a nonempty edge on ωi or a vertex with valency at least ﬁve in Li+1 . Then by Proposition 1.20, M has transversals α and β starting at the endpoints of ωi , respectively, such that Mα ∩Mβ = ∅. But then it follows from Proposition 1.21 that Mα and Mβ contain disjoint peaks Pα and Pβ , respectively. Since uμvν −1 is a boundary cycle of M , this implies that either ∂Pα ∩ ∂M ⊆ μ or ∂Pβ ∩ ∂M ⊆ ν, where Mα is to the left of Mβ . This however violates the assumption, that μ contains no peaks. Therefore, every vertex of M has valency at most ﬁve and every layer Li of M contains at most four regions, which have a common edge with ωi . We compute the number of regions in Li+1 . Let a be the number of regions of Li+1 which have a common edge with ωi and let b be the number of regions of Li+1 which have only a common vertex with ωi . Then |Li+1 | = a + b. Now, if ωi = w0 θ1 w1 θ2 · · · wr θr wr+1 , where wj vertices and θi edges, then r = a and each vertex contributes at most d(w) − 3 = 1 to b, if w = w0 and w = wr and w0 and wr contribute at most d(w) − 2 = 2. Consequently, b ≤ 1 · (r − 1) + 2 · 2 ≤ r + 3. Since r ≤ 3, |Li+1 | = a + b ≤ 3 + 3 + 3 = 9. Hence, for every layer Li of Λ, we have |μi | ≤ 9|R| and |νi | ≤ 9|R|. Together with (1.1) we get 1 ≤ |μi | ≤ 9|R| and 1 ≤ |νi | ≤ 9|R|.

(1.3)

186

A. Juh´ asz But then by (1.2) and (1.3) we have s ≤ |μ| ≤ 9s|R| and s ≤ |ν| ≤ 9s|R|. Consequently, |ν| ≤ 9s|R| ≤ 9|μ||R| = 9|U ||R|. |ν| i Now, denote S = S . Then S is a set of words computable from S i=0

since |ν| < c|μ| and c|μ| is given. Let S0 the subset of all the elements of S which are equal to U in G. Since G has solvable word problem due to the condition C (1/5) & T (4), S0 is computable from S. Finally, let S1 be the subset of S0 of the elements of H in S1 . By assumption, S1 is non-empty, hence S1 can be constructed from S. The proposition is proved.

Completion of the proof. Let U be a reduced word in F , U = 1 with θ(U ) ∈ θ(H). Then there exists a reduced word V ∈ H with θ(U V −1 ) = 1. Therefore, by Theorem 1.1 there exists a van Kampen diagram M with boundary cycle uμvν −1 with Φ(μ) = U and Φ(ν) = V , where u and v are vertices. Since the word problem for G is solvable, we may assume that U is a shortest element among the representatives of θ(U ). Hence Proposition 1.25 applies. Applying Proposition 1.25 it follows that |V | ≤ 9|U ||R| and V can be written down eﬀectively, knowing U . This solves the Membership Problem for H. The theorem is proved.

2. Word combinatorics The aim of this section is to introduce the basic results in word combinatorics which enable us to show in Section 3 that P is almost σ-complete. 2.1. Words Let F be a free group, freely generated by a set X and let W be a reduced word in F . Denote by H(W ) the set of initial subwords of W and by T (W ) the set of terminal subwords of W . Also, for a reduced non-empty word W we denote by h(W ) the ﬁrst letter of W and by t(W ) the last letter of W . Denote by Supp(W ) the set of all the elements of X which occur in W or W −1 . We have the following well-known result. Lemma 2.1. Let A, B and C be reduced words such that AB and BC are reduced as written. If AB = BC then A = KL, C = LK and B = (KL)β K, β ≥ 0. We introduce below the key notion of the work. Deﬁnitions and notations. (a) Let W1 and W2 be reduce words in F . W2 majorises W1 if Supp(W2 ) ⊇ Supp(W1 ). In this case write W2 W1 . If W1 W2 and also W1 W3 we shall write W1 W2 , W3 . Also, if Supp(W1 ) ∪ Supp(W2 ) ⊇ Supp(W3 ) we shall write W1 , W2 W3 .

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(b) For W1 and W2 in part (b) deﬁne W1 ∼ W2 if W1 ≺ W2 and W2 ≺ W1 . Thus W1 ∼ W2 if and only if Supp(W1 ) = Supp(W2 ). Clearly “∼” is an equivalence relation, which contains the equality of elements in F . The following lemma is immediate from the deﬁnition, hence its proof is omitted. Lemma 2.2. (a) (b) (c) (d) (e)

If A is a subword of B then A ≺ B. If A ≺ B then A±1 ≺ B ±1 . If A ∼ B and A ≺ C then B ≺ C. If A = P1 . . . Pm , reduced as written and Pi ∼ Q for i = 1, . . . , m then A ∼ Q. If A P1 , . . . , Pm then A W (P1 , . . . , Pm ), for every word W on P1 , . . . , Pm .

Parts (a) and (b) of the following lemma are immediate corollaries of Lemma 2.1 and Lemma 2.2, hence we omit their proofs. Lemma 2.3. (a) Let A, B and C be as in Lemma 2.1 (a). Then B ≺ A ∼ C ∼ AB ∼ BC. If β ≥ 1 then B ∼ A. (b) If AB = KAC with AB and KAC, reduced as written then B A, B C and K A. (c) Let K, Q, U, V and S be non-empty words such that KQ, U V, V U and KS are reduced as written, of length at least two. If KQ = U V and KS = V U then Q ∼ S K, U, V . (d) Let B, Q, L, U and V be non-empty words such that BQ, U V, LB and V U are reduced as written. If BQ = U V and LB = V U then one of the following holds: (i) B = U, Q = L = V or (ii) Q B, U, V, L and L B, U, V, Q (hence L ∼ Q ∼ U V ). (e) Let L, K, Q1 , M and N be non-empty reduced words, such that KQ1 , M N , Q1 M and LK are reduced as written with length at least two. If KQ1 = M N and Q1 M = LK, then one of the following holds: (i) Q1 = N = L and K = M or (ii) Q1 K, L, M, N . Proof. We prove here only part (c), because the proofs of parts (d) and (e) follow the same line of proof. We prove part (c) by induction on |U V |. If |U V | = 2 then K = U = S and Q = V = K, hence the claim of part (c) of the lemma holds true. Assume |U V | > 2 and consider the ﬁrst equation, KQ = U V . Then one of the following holds: Case 1. K = U . Then Q = V . Substitution in the second equation gives U S = V U , hence the result follows by part (a).

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Case 2. K = U K1 , K1 = 1. Then V = K1 Q, hence U K1 S = K1 QU . By Lemma 2.1 K1 S = M N , K1 Q = N M , U = (M N )u M , u ≥ 0. Since |M N | < |U V | and K1 = 1, we may apply the induction hypothesis on these equations to give S K1 , M, N, Q K1 , M, N and Q ∼ S. Consequently tracing back K and V we get S K, U, V . Case 3. U = KU1 , U1 = 1. Then Q = U1 V . Substitution in the second equation yields KS = V KU1 . The result follows by part (b) and Lemma 2.2.

The lemma is proved.

Recall from Deﬁnition 1.22 that σ(W ) denotes the set of all the subwords of W with length two. We need some further deﬁnitions. Deﬁnitions 2.4. Let W be a reduced word. (a) A decomposition of W is a partition of W into subwords. More precisely, a decomposition of W into k parts (subwords) is a function δk : F → (F )k := F × · · · × F such that if εk : (F )k → F is the function εk (x1 , . . . , xk ) = k times

x1 · · · xk for xi ∈ F , then εk ◦ δk (W ) = W . If δk (W ) = (W1 , . . . , Wk ) then we shall write this by W = W1 ∗ W2 ∗ · · · ∗ Wk . By deﬁnition, W1 · · · Wk = W , ∞ k (F ) and δ : F → F∞ . We shall use reduced as written. Denote F∞ = k=1

this notation when k is not important. (b) Let δ : W = W1∗ · · · ∗ Wk be a decomposition of W. Deﬁne τδ (W ) = t(Wi )h(Wi+1 ) i = 1, . . . , k − 1 . Thus τδ (W ) is a subset of σ(W ) which contains at most k − 1 elements. When δ is clear from the context we shall write τ (W ) for τδ (W ). (c) We shall need to consider cyclic words. Assume W is cyclically reduced the cyclic word which corresponds to W . Then σ(W ) = and denote by W 2 σ(W ) = σ(W ) ∪ {t(W )h(W )} and if W = W1 ∗ · · · ∗ Wk then τ (W ) = ) by σ ) by τ (W ) ∪ {t(Wk ) · h(W1 )}. We shall also denote σ(W (W ) and τ (W τ(W ), when convenient. The following two lemmas are immediate from the deﬁnition. Lemma 2.5. Let W be a reduced word in F and let δ : W = W1 ∗ · · · ∗ Wk , k ≥ 2, a decomposition of W . Then 0 / k σ(Wi ) ∪ τδ (W ). (a) σ(W ) = i=1

) = σ(W ) ∪ {t(W )h(W )}. (b) If W is cyclically reduced then σ(W Lemma 2.6. Let A and B be reduced non-empty words. (a) If A is a subword of B then σ(A) ⊆ σ(B). (b) If A1 ∈ T (A) and B0 ∈ H(B) non-empty words, then τ (A ∗ B) = τ (A1 ∗ B0 ).

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(c) Let in addition C be a word with decomposition C1 ∗ · · · ∗ Ck . ≥ |σ(B)| − k. If σ(A) = σ(B) ∪ τ (C1 ∗ · · · ∗ Ck ) then |σ(A)| Lemma 2.7. Let AB = BC reduced as written, A = 1, B = 1. Then (a) σ(ABC) = σ(AB) = σ(BC). (b) σ(AB) = σ(A) ∪ τ (A ∗ B), σ(BC) = σ(C) ∪ τ (B ∗ C). Proof. (a) Clearly, σ(AB) ⊆ σ(ABC) and σ(BC) ⊆ σ(ABC), by Lemma 2.6(a). We show that σ(ABC) ⊆ σ(AB). σ(ABC)

= by Lemma 2.5(a)

σ(AB) ∪ τ (AB ∗ C) σ(AB) ∪ τ (B ∗ C)

=

by Lemma 2.6(b)

⊆

by Lemma 2.5(a)

=

since AB=BC

σ(AB) ∪ σ(BC) σ(AB) = σ(BC).

(b) We have A = KL, B = (KL)b K, b ≥ 0 and C = LK by Lemma 2.1. Hence σ(AB) = σ(A)∪σ(B)∪τ (A∗B) by Lemma 2.5(a). But σ(B) = σ (KL)b K ⊆ σ(KL) ∪ τ (L ∗ K) = σ(A) ∪ τ (L ∗ K). Therefore, σ(AB) ⊆ σ(A) ∪ τ (L ∗ K).

(2.4)

Since LK and A are subwords of AB, hence by Lemma 2.6(a) σ(A) ∪ τ (L ∗ K) ⊆ σ(AB).

(2.5)

From (2.4) and (2.5) we get σ(AB) = σ(A) ∪ τ (L ∗ K).

(2.6)

By Lemma 2.6(b) we have τ (L ∗ K) = τ (A ∗ B), since L ∈ T (A) and K ∈ H(B). Substituting this in (2.6) gives σ(AB) = σ(A) ∪ τ (A ∗ B). The remaining equation of part (b) follows by the same argument. The lemma is proved. Corollary 2.8. Suppose W = A ∗ B ∗ C. If AB = BC then σ(W ) = σ(A) ∪ τ (A ∗ B) and also σ(W ) = σ(C) ∪ τ (B ∗ C). Finally, we need the following basic notions. Deﬁnition 2.9. (a) Let R be a cyclically reduced word in F and let P be a subword of a cyclic conjugate of R. P is a piece in R (or a piece relative to the symmetric closure R of R) if R has distinct cyclic conjugates R 1 and R2 such that R1 = P R1 , R2ε = P R2 , reduced as written, for some ε ∈ {1, −1}. We call the two occurrences of P in R1 and R2ε , respectively, a piece-pair and denote it by (P, P ), where P = P ε is the occurrence of P ε in R2 . We shall deliberately use both notations ε(Pi ) and εi for ε, if there are several pieces Pi , as convenient.

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(b) A piece pair (P, P ) as in part (a) of the deﬁnition is right normalized if R1−1 R2 is reduced as written. Let R1 , R2 ∈ R. Then by deﬁnition R2 is a cyclic conjugate of R1ε , for some ε ∈ {1, −1}. We can express this fact by the following two word equations: R1 = KL and R2ε = LK for some words L and K, reduced as written. Hence the fact that P is a piece, as in Deﬁnition 2.9, we can express via the following set of equations: P R1 = KL and P R2 = (LK)ε . More generally, if D is a region in M with adjacent neighbours E1 and E2 then we can write R = P1 P2 Q, where P1 = Φ (∂D ∩ ∂E1 ), P2 = Φ (∂D ∩ ∂E2 ). Clearly, P1 and P2 are pieces, hence as above we can write for them a system of word equations. In Section 3 we are going to consider this system of word equations, together with the following two equations U XP1 = R1 and P2 V Y = R2 , where X and Y are pieces and R1 and R2 cyclic conjugates of R± . We are not interested in solving this equations, but rather use them as carriers of the set theoretical information we need, namely σ(X), σ(Y ), Supp(X) → S, where Fact(R) is the set of all the subwords and Supp(Y ). Thus, σ : Fact(R) and we of R and S is the set of all the subsets of subwords of length two of R would like to show that σ(Y QX) contains every subword of R of length two, with the exception of at most three. The point is that this is a set theoretical statement which is much easier to prove that solve ﬁrst the set of equation and then for each solution to show that σ(Y QX) is as written above.

3. Piece conﬁgurations of 1-corner regions and 2-corner regions In this section we assume the conditions of the Main Theorem are satisﬁed. 3.1. 1-corner regions Let D be a 1-corner region in M with inner neighbour E. Denote α = ∂D ∩ ∂E, see Fig. 8(a). P is a let P = Φ(α) and let Q = Φ(∂D ∩ ∂M ). Thus, σ(Q) ⊆ σ(R), piece and vP uQ is a boundary label of D with vertices u, v.

(a)

(b) Figure 8

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− 2. Lemma 3.1. Let notation be as above. Then Q P and σ(Q) ≥ σ(R) Proof. Let (P, P ) be the corresponding piece pair. Then one of the following holds: (1) P is a subword of Q; (2) P contains u as an inner vertex; (3) P contains v as an inner vertex. In case (1) Q P , by Lemma 2.2 (a). In cases (2) and (3) we assume that |P | ≥ 2. In case (2) we have P = AX, P = XY, Q = Y Q1 , reduced as written, Q1 ∈ T (Q). See Fig. 8(b). Applying Lemma 2.3(a) to the ﬁrst two of these equations and remembering that P −1 cannot overlap P , we get A ∼ Y X and hence P ∼ Y , by Lemma 2.2(d). Applying Lemma 2.2(a) to the last equation implies Q P . Finally, Case 3 is dual to Case 2. Hence in all the cases Q P . Also, it follows from Lemma 2.7(b) that σ(P ) = σ(XY ) = σ(Y ) ∪ τ (X ∗ Y ).

(3.7)

Since X ∈ T (P ) and Y ∈ H(Q), by Lemma 2.6(b), τ (X ∗ Y ) = τ (P ∗ Q).

(3.8)

Since Y is a subword of Q, by Lemma 2.6(a), σ(Y ) ⊆ σ(Q).

(3.9)

Combining (3.8) and (3.9) with (3.7) we get σ(P ) ⊆ σ(Q) ∪ τ (P ∗ Q).

(3.10)

Now, σ(R)

=

σ(P ) ∪ σ(Q) ∪ τ (P ∗ Q) ∪ τ (Q ∗ P )

=

σ(Q) ∪ τ (P ∗ Q) ∪ τ (Q ∗ P ).

by Lemma 2.5 by (3.10)

≥ |σ(Q)| − 2, as required. Therefore, by Lemma 2.6(c) σ(R) The lemma is proved.

3.2. 2-corner regions Let D be a 2-corner region in M with inner neighbours Er and E . See Fig. 9. Denote α1 = ∂D ∩ ∂Er and denote α2 = ∂D ∩ ∂E . Let v0 = α1 ∩ ∂M , let v2 = α2 ∩ ∂M and let v1 = α1 ∩ α2 . Denote P1 = Φ(α1 ), P2 = Φ(α2 ) and Q = Φ(∂D ∩ ∂M ). It is convenient and harmless to identify Pi with αi and, similarly, Pi with αi , i = 1, 2. Then v2 Qv0 P1 v1 P2 v2 is a boundary label of D, which we may assume to coincide with R, without loss of generality, hence P1 and P2 are pieces. Let (P1 , P1 ) and (P2 , P2 ) be the corresponding piece pairs. Then P1 and P2 are subwords of the cyclic word R or R−1 , hence one of the following holds for each of P1 and P2 : Case 1 v0 is an inner vertex of P1

Case 1 v2 is an inner vertex of P2

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A. Juh´ asz Case 2 v1 is an inner vertex of P1

Case 2 v1 is an inner vertex of P2

Case 3 v2 is an inner vertex of P1

Case 3 v0 is an inner vertex of P2

Case 4 P1 is a subword of P2

Case 4 P2 is a subword of P1

Case 5 P1 is a subword of Q

Case 5 P2 is a subword of Q

Figure 9 We propose to show that in most of the cases Q P1 P2 . (See precise statement below.) We see that for i = 1, 2, 3, 4 and 5, Case i is the dual of Case i obtained by exchanging P1 with P2 and v0 with v2 . Hence out of the 25 cases (i, j ), 1 ≤ j , i ≤ 5, only those with j ≥ i have to be checked, because the rest is obtained by duality. The following is the main result of this section. Proposition 3.2. Let the notation be as above and assume that R satisﬁes the assumptions of the Main Theorem. Assume that the piece pairs (P1 , P1 ) and (P2 , P2 ) are right normalised. Let Qr = ∂Er ∩ ∂M and let Q = ∂E ∩ ∂M . Then σ(Qr ), and one of the following holds: σ(Q ) and σ(Q) are subsets of σ(R) − 3. (a) Q P1 P2 and |σ(Q)| ≥ σ(R) (b) If both v0 and v2 have valency three and both Qr and Q are not pieces (i.e., the products of at least two pieces) then either Qλ Q R or QQρ R. Also − 3 or |σ(QQρ )| ≥ σ(R) − 3. either |σ(Qλ Q)| ≥ σ(R) (c) Moreover, Q P1 in cases (1, j ), j = 1, 2, 3, 4, 5 and Qρ P1 in cases (2, j ), j = 2, 3, 4, 5; Q P1 , P2 in cases (3, j), j = 3, 4, 5 and cases (4, 5) and (5, 5); Qρ P1 , P2 and Qλ P1 , P2 in case (4, 4). Dually, Qλ P2 in cases (i, 1), i = 1, 2, 3, 4, 5 and cases (i, 2), i = 2, 3, 4, 5 Q P1 , P2 in cases (i, 3), i = 3, 4, 5 and case (5, 4).

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Proof. As explained above it is enough to check the 15 cases (i, j), 1 ≤ i ≤ j ≤ 5. We shall check here only the case (i, j ) = (1, 1). The rest can be completed by borrowing the results from [J2, Section 2.2, pp. 183–189], together with Lemmas 2.5–2.8. For the sake of completeness we present the details in the Appendix. Case (1, 1 ). Since P1 and P1 overlap and F contains no elements of order two hence ε(P1 ) = 1. Therefore we have P1 = Q1 X, Q1 ∈ τ (Q) and P1 = XY . Hence P1 = Q1 X = XY and by Lemma 2.3 Q1 ∼ X ∼ Y . Therefore by Lemma 2.2 Q Q1 P1 and P1 ≺ Q. Also, Q1 P1 = Q1 XY , hence σ(Q1 P1 ) = σ(Q1 ) ∪ τ (Q1 ∗ P1 ), by Lemma 2.6(b) and Corollary 2.8. Hence we have: P1 ≺ Q

and σ(Q1 P1 ) = σ(Q1 ) ∪ τ (Q1 ∗ P1 ).

(3.11)

We show that (QP1 P2 ) ≥ |σ(Q)| − 3. P2 ≺ Q and σ

(3.12)

This follows from the above argument with P1 and P2 interchanged hence P2 ≺ Q follows from (3.11). It also follows from (3.11) that σ(P2 Q0 ) = σ(Q0 ) ∪ τ (P2 ∗ Q0 )

(3.13)

for an initial subword Q0 of Q such that Q = Q0 Q2 Q1 . Hence, by Lemma 2.5 σ(R) = σ(QP1 P2 ) = σ(Q) ∪ σ(P1 ) ∪ σ(P2 ) ∪ τ(Q ∗ P1 ∗ P2 ) =

σ(Q) ∪ σ(Q0 ) ∪ σ(Q1 ) ∪ τ(Q ∗ P1 ∗ P2 )

=

∪ τ (P2 ∗ Q0 ) ∪ τ (Q1 ∗ P1 ) σ(Q) ∪ τ(Q ∗ P1 ∗ P2 ) ∪ τ (P2 ∗ Q0 ) ∪

by (3.11) and (3.13)

by Lemma 2.6(a)

∪ τ (Q1 ∗ P1 ). But by Lemma 2.6(b) τ (P2 ∗ Q0 ) = τ (P2 ∗ Q) and τ (Q1 ∗ P1 ) = τ (Q ∗ P1 ). Hence, τ(Q ∗ P1 ∗ P2 ) ∪ τ (P2 ∗ Q0 ) ∪ τ (Q1 ∗ P2 ) = τ(Q ∗ P1 ∗ P2 ). Consequently, = σ(Q)∪ τ(Q∗P1 ∗P2 ) and by Lemma 2.6(c) | σ (R)| ≥ |σ(Q)|−3, as required. σ(R) We close this section with the following consequence of the proposition. Proposition 3.3. Let M be an R-diagram. Let P be a peak relative to a layer decomposition Λ with Lp = L1 . Let α = ∂P ∩ ∂M . Then Supp Φ(α) ⊇ Supp(R) and |σ Φ(μ) | ≥ |R| − 3. Proof. Let P = D1 , . . . , Dk . If k = 1 then the result follows from Lemma 3.1. If k ≥ 3 then it follows from Theorem 1.7(d) and the T (4) condition that either P contains a 1-corner region or a 2-corner region D with two neighbours Er and E such that ∂D ∩ ∂Er ∩ ∂M and ∂D ∩ ∂E ∩ ∂M are vertices with valency three and ∂Er ∩ ∂M and ∂E ∩ ∂M are not pieces (due to the C (1/5)-condition). See Fig. 6(a), where D1 is E , D2 is D and D3 is Er . In both cases the result follows by part (b) of Proposition 3.2. Hence, if Int(Lp ) has a component P such that |P | = 2

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then we are done. Therefore, assume that every peak in Lp contains exactly two regions. We show that then (∂P ∩ ∂M )∪(∂L p−1 ∩ ∂M ) contains a subpath μ such that Supp Φ(μ) ⊇ Supp(R) and σ Φ(μ) ≥ | σ (R)| − 3. Let P = D1 , D2 be an extremal peak, say left extremal. (See Fig. 11(b).) Then both D1 and D2 are 2-corner regions, and D1 has one neighbour E in Lp−1 . Since D1 is a left extremal region in Lp , D1 is the only neighbour of E1 in Lp and dM (E) ≤ 3 + 1 = 4. Due to the C(6)-condition ∂E ∩ ∂M is the product of at least two pieces and since D1 is left extremal the vertex v := ∂D1 ∩∂E ∩∂M has valency three. It follows from Theorem 1.7(d) that w := ∂D1 ∩ ∂D2 ∩ ∂M also has valency three in M . Therefore Proposition 3.2(b) applies and itimplies that ∂Lp ∩ ∂M contains a subpath μ with Supp Φ(μ) ⊇ Supp(R) and σ Φ(μ) ≥ | σ (R)| − 3, as required. The proposition is proved. Proposition 3.4. Let M be a connected simply connected R diagram with connected interior. Let Λ be a layer decomposition of M with center v, where v is a boundary Then ∂M has a subpath μ with v ∈ / μ such that Supp Φ(μ) ⊇ Supp(R) and vertex. σ Φ(μ) ≥ | σ (R)|−3. In other words, if P = X|R then P is almost σ-complete. Proof. Suppose ﬁrst that M consists of a single region and let ω a boundary cycle with o(ω) = t(ω) = v. Then clearly Supp Φ(ω) ⊇ Supp(R) and σ Φ(ω) ≥ | σ (R)| − 1. So assume M consists of more than one region. If M = L1 (v) then v is a common boundary vertex of all the regions of M by deﬁnition of L1 (v). Therefore either M contains a (boundary) region E with dM (E) = 1 or it contains at least three regions each of which has at most two neighbours and one of which satisﬁes the conditions of Proposition 3.2(b). In this case the result follows by p Proposition 3.2(b). Finally assume M = Li , p ≥ 2. Then the statement of the i=0

proposition follows by Proposition 3.3. The proposition is proved.

3.3. Proof of the Main Theorem By Theorem 1.24 it is enough to show that P is almost σ-complete. But P is almost σ-complete by Proposition 3.4. The theorem is proved.

Appendix Case 2 . Consider two subcases according as P2 is a subword of P1 P2 or is not a subword of P1 P2 . In both cases P2 and P2 overlap, hence ε(P2 ) = 1. Subcase 1. P2 is a subword of P1 P2 . We have P1 = U X, P2 = XY = Y V . Consequently, P1 X, X ∼ V ∼ P2 by Lemma 2.3 and hence P1 ≺ P2 by Lemma 2.2. Hence (3.11) implies Q P2 . Also

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σ(P2 ) = σ(X) ∪ τ (X ∗ Y ) by Lemma 2.7(b). Since X ∈ T (P1 ) and Y ∈ H(P2 ), hence τ (X ∗Y ) = τ (P1 ∗P2 ) by Lemma 2.6(b) and hence σ(P2 ) = σ(X)∪τ (P1 ∗P2 ). Since X is a subword of P1 , hence σ(X) ⊆ σ(P1 ) by Lemma 2.6(a), and therefore σ(P2 ) ⊆ σ(P1 ) ∪ τ (P1 ∗ P2 ). Combining this with (3.11) and using Lemma 2.5 gives σ (QP1 P2 ) = σ(Q)∪ τ (Q∗P1 ∗P2 ), hence | σ (QP1 P2 )| ≥ |σ(Q)|−3 by Lemma 2.6(c), as required. Subcase 2. P2 is not a subword of P1 P2 . We have P2 = Q2 P1 X, Q2 ∈ τ (Q), P2 = XY . Hence (Q2 P1 )X = XY and Q2 P1 ∼ Y ∼ P2 . But Q2 ≺ Q by Lemma 2.2, and P1 ≺ Q by (3.11), hence P2 ∼ Q2 P1 implies that P2 ≺ Q. Also, σ(P2 ) = σ(XY ) = σ(Q2 P1 ) ∪ τ (Q2 P1 ∗ X) by Lemma 2.7(b). Since X ∈ H(P2 ), hence τ (Q2 P1 ∗X) = τ (P1 ∗P2 ), by Lemma 2.7 and hence (3.14) σ(P2 ) = σ(Q2 P1 ) ∪ τ (P1 ∗ P2 ). Since Q1 and Q2 are subwords of Q it follows from (3.11) that σ(QP1 ) = σ(Q) ∪ τ (Q ∗ P1 ) and from (3.14) that σ(P2 ) ⊆ σ(QP1 ) ∪ τ (P1 ∗ P2 ). Hence σ (QP1 P2 ) = σ(Q) ∪ τ (Q ∗ P1 ) ∪ τ (P1 ∗ P2 ) ∪ τ (P2 ∗ Q), by Lemma 2.5. But then σ (QP1 P2 ) ≥ |σ(Q)| − 3, by Lemma 2.6(c). Case 3 . We may assume that v1 ∈ / P2 , by Subcase 2 above. Then P2 = Q2 X, Q2 ∈ τ (Q) and P1 = XY . Therefore Q2 ≺ Q and X ≺ Q due to (3.11). Consequently, P2 = Q2 X ≺ Q, by Lemma 2.2 and (3.12) follows. Also, since P2 = Q2 X, σ(P2 ) = σ(Q2 ) ∪ σ(X) ∪ τ (Q2 ∗ X). Since Q2 ∈ T (Q) and X ∈ H(P1 ), we have σ(P2 ) ⊆ σ(Q) ∪ σ(X) ∪ τ (Q ∗ P1 ). But since X is a subword of P1 , σ(X) ⊆ σ(P1 ), hence σ(P1 ) ∪ σ(P2 ) ∪ σ(Q) ⊆ σ(P1 ) ∪ σ(Q) ∪ τ (Q ∗ P1 ). Hence due to (3.11) we get σ (P1 P2 Q) = σ(Q) ∪ τ(P1 ∗ P2 ∗ Q) and by Lemma 2.6(c) the result follows. Case 4 . If ε(P2 ) = 1 then P2 ≺ P1 , hence P2 ≺ Q, due to (3.11) and Lemma 2.2. If ε(P2 ) = −1 then P2−1 ≺ P1 , hence we have P1 ≺ Q and P2−1 ≺ Q. Also due to (3.11) we get σ (P1 P2 Q) = σ(Q) ∪ τ (P1 ∗ P2 ∗ Q), and hence the result follows by Lemma 2.7(c). Case 5 . Similar to Case 4 . Case (2, 2 ). Consider 4 subcases according to whether v2 belongs or does not belong to P1 and dually, v0 belongs or does not belong to P2 . / P1 and v0 ∈ / P2 . (See Fig. 10(a).) Subcase 1. v2 ∈ Then P1 = XY, P1 = Y Z, P2 = ZT , hence τ (Y ∗ Z) ∈ σ(P1 ) and σ(P1 ) = σ(X) ∪ τ (X ∗ Y ). Now, τ (Y ∗ Z) = τ (P1 ∗ P2 ), by Lemma 2.6(b) since Y ∈ T (P1 ) and Z = H(P2 ), hence τ (P1 ∗ P2 ) ⊆ σ(P1 ). (3.15)

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(a)

(b)

(c) Figure 10 Suppose dM (v0 ) = 3. Fig. 10(a) shows the relative positions of P1 and P1 on ∂D, due to the assumption that v1 ∈ P1 . (P1 has an occurrence on ∂D because Er and D have the same boundary labels.) We see that in order to get P1 from P1 we have to turn P1 along ∂D anticlockwise by |X|. Since D and Er have the same boundary labels (i.e., labels of boundary cycles and their inverses) we can reproduce P1 on ∂Er by turning P1 clockwise along ∂Er by |X|. But P1 starts at v0 on ∂Er just as P1 starts at v0 on ∂D. Hence Qr has X −1 as a head, provided that |Qr | ≥ |X|. But X is a piece while Qr is not, hence X is a proper subword of Qr . We call this occurrence of P1 on ∂Er the image of P1 on ∂Er and denote it by P 1 . Hence X −1 ∈ H(Qr ). Deﬁne Qρ = X. Then we get P1 = Qρ Y = Y Z, P2 = ZT . Therefore σ(P1 ) = σ(Qρ ) ∪ τ (X ∗ Y ). Since by (3.15) τ (P1 ∗ P2 ) ⊆ σ(P1 ), we get σ(P1 P2 ) = σ(P1 ) ∪ σ(P2 ) ∪ τ (P1 ∗ P2 ) = σ(P1 ) ∪ σ(P2 ), hence σ(P1 P2 ) = σ(Qρ ) ∪ σ(P2 ) ∪ τ (X ∗ Y ).

(3.16)

Since ε(P1 ) = −1 hence Qρ Y = Z −1 Y , hence by Lemma 2.1(b) |Y | = 1 and Qρ ∼ Z −1 Y −1 = P1−1 . Hence Qρ P1 . So assume ε(P1 ) = 1. Using Lemma 2.3(a), the ﬁrst pair of equations gives Qρ ∼ P1 .

(3.17)

But P2 and P2 overlap, since v1 is an inner vertex of P2 , hence P2 = LV, P2 = KL and P1 = U K. Applying Lemma 2.3(a) to the ﬁrst pair of equations, if |P2 | ≥ 2 and ε(P2 ) = 1, then P2 ∼ K,

(3.18)

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while the last equation gives K ≺ P1 .

(3.19)

Now, from (3.17), (3.18) and (3.19) we get P1 ∼ Qρ , P2 ≺ Qρ . Also σ(P2 ) = σ(K)∪τ (K ∗L), by Lemma 2.7. Since K ∈ T (P1 ) and L ∈ H(P2 ), hence τ (K ∗L) = τ (P1 ∗ P2 ), by Lemma 2.6(b) and hence σ(P2 ) ⊆ σ(P1 ) ∪ τ (P1 ∗ P2 ). Thus, (3.16) implies σ(P1 P2 ) = σ(Qρ ) ∪ τ (X ∗ Y ). Consequently, σ (QP1 P2 ) = σ(Q) ∪ σ(P1 P2 ) ∪ τ (P2 ∗ Q) ∪ τ (Q ∗ P1 ) ⊆ σ(Q) ∪ σ(Qρ ) ∪ τ (X ∗ Y ) ∪ τ (P2 ∗ Q) ∪ τ (Q ∗ P1 ). Therefore, |σ(Q) ∪ σ(Qρ )| ≥ | σ (QP1 P2 )| − 3 and since σ(QQρ ) = σ(Q) ∪ σ(Qρ ) ∪ τ (Q ∗ Qρ ), hence σ(QQρ ) ≥ | σ (QP1 P2 )| − 3. / P1 and v0 ∈ P2 . (See Fig. 10(b).) Subcase 2. v2 ∈ Then (3.17) above still holds, and for P2 and P2 we get P2 = HP1 L, P2 = LK, where H ∈ T (Q). Hence, by Lemma 2.3(a), HP1 ∼ P2 , i.e., QQρ P2 , P1 . Also, P2 = LK = HP1 L implies by Lemma 2.7 that σ(P2 ) = σ(HP1 )∪τ (P1 ∗L) and since L ∈ H(P2 ), hence τ (P1 ∗ L) = τ (P1 ∗ P2 ). Hence, σ(P2 ) = σ(HP1 ) ∪ τ (P1 ∗ P2 ) = σ(H) ∪ σ(P1 ) ∪ τ (H ∗ P1 ) ∪ τ (P1 ∗ P2 ), by Lemma 2.5(a). Now, H ∈ T (Q), hence, τ (H ∗ P1 ) = τ (Q ∗ P1 ) and hence σ(P2 ) ⊆ σ(QP1 ) ∪ τ (P1 ∗ P2 ). Consequently, σ (QP1 P2 )

=

σ(Q) ∪ σ(P1 ) ∪ σ(P2 ) ∪ τ(Q ∗ P1 ∗ P2 )

=

σ(Q) ∪ σ(P1 ) ∪ τ (Q ∗ P1 ) ∪ τ (P1 ∗ P2 ) ∪

=

∪ τ (P2 ∗ Q) σ(Q) ∪ σ(Qρ ) ∪

due to (3.17)

∪ τ(Q ∗ P1 ∗ P2 ). σ (QP1 P2 )| − 3. As above, this implies |σ(QQ−1 ρ )| ≥ | Subcases 3 and 4. v2 ∈ P1 . (See Fig. 10(c).) Then P1 = XY, P1 = Y P2 T where T ∈ H(Q). Therefore, σ(P1 P2 T ) = σ(XY P2 T ) = σ(X) ∪ τ (X ∗ Y ), by Lemma 2.7. Consequently, σ (P1 P2 Q) = =

σ(X) ∪ σ(Q) ∪ τ (X ∗ Y ) ∪ τ (Q ∗ P1 ) ∪ τ (P2 ∗ Q) σ(Qρ ) ∪ σ(Q) ∪ τ (X ∗ Y ) ∪ τ (Q ∗ P1 ) ∪ τ (P2 ∗ Q)

σ (QP1 P2 )|−3. Since ε(P1 ) = 1, by Lemma 2.3(a), and consequently, |σ(QQ−1 ρ )| ≥ | X ∼ P1 ∼ P2 T Now, X ∈ H(Qr ) as in Subcase 1. Thus P1 , P2 ≺ Qr . Case (2, 3). v1 ∈ P1 and v0 ∈ P2 . (See Fig. 11(a) and Fig. 11(b).) We consider three main cases according as v1 ∈ / P2 and v2 ∈ / P1 or v1 ∈ P2 or v1 ∈ / P2 and v2 ∈ P1 . In each main case we consider two cases according as ε2 = 1 or ε2 = −1.

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(b) Figure 11

Main Case 1. v1 ∈ / P2 and v2 ∈ / P1 . We have P2 = Q1 U, P1 = U V, P1 = XY, P1 = Y Z and P2 = ZW, h(W ) = h(Z)

(3.20)

where all subwords Q1 , U, V, X, Y, Z and W are non-empty, Q1 ∈ τ (Q) and P1 , P2 , P1 and P2 are reduced as written. Also, h(W ) = h(Z) because the piecepair (P1 , P1 ) is right normalized. Since F contains no elements of order two, may assume ε1 = 1. Then P2 = Q1 U = ZW

and P1 = XY = Y Z = U V.

(3.20 )

Observe that Q1 ∈ T (Q)

and X −1 ∈ H(Qr ).

(3.20 )

It follows from (3.20 ) and Lemma 2.1 that X = KL, Y = (KL)k K, k ≥ 0, Z = LK.

(3.20 )

It follows from (3.20 ) and Lemma 2.3(i) that X Y, Z and by the second and fourth equations in (3.20 ) also X U, V . Hence if we deﬁne Qρ = X −1 then Qρ Y, Z, U, V . Since P2 = Q1 U hence QQρ Q1 , Qρ and hence QQρ P2 , P1 . Also, it follows from (3.20 ) that σ(P1 ) = σ(X) ∪ τ (X ∗ Y ) and σ(P2 ) = σ(Q1 ) ∪ τ (Q1 ∗ U ). Therefore, σ (QP1 P2 ) = σ(Q) ∪ σ(X) ∪ τ (X ∗ Y ) ∪ τ (Q1 ∗ U ) ∪ τ (Q ∗ P1 ) ∪ τ (P1 ∗ P2 ) ∪ τ (P2 ∗ Q) . But τ (Q1 ∗ U ) = τ (Q ∗ P1 ) and τ (P1 ∗ P2 ) = τ (Y ∗ Z) ∈ σ(P1), by (3.20 ) hence σ (QP1 P2 ) = σ(Q) ∪ σ(X) ∪ τ (X ∗ Y ) ∪ τ (Q ∗ P1 ) ∪ τ (P2 ∗ Q) . −1 ∈ H(Qr ), by (3.20 ), we get |σ(Q) ∪ σ(Qr )| ≥ | σ (QP1 P2 )| − 3, by Since X Lemma 2.6(c). Main Case 2. v1 ∈ P2 . Then P2 = Q1 P1 U, P2 = U V, Q1 ∈ τ (Q), all expressions reduced as written. Assume that ε2 = −1. Then P2 = U −1 P1−1 Q−1 1 = U V , hence by Lemma 2.1(b)(i) |U | = 1 and P2 ∼ P1−1 Q−1 . Therefore QQ Q1 Qρ P1 , P2 . ρ 1 May assume ε2 = 1. Then P2 = Q1 P1 U, P2 = U V . Consequently, v2 ∈ / P1 since |P2 | > |P1 | and hence P1 = XY, P1 = Y Z and P2 = ZW . The equations P2 = U V = Q1 P1 U imply that V ∼ P2 ∼ Q1 P1 , by Lemma 2.3. Also, we observe that V −1 ∈ τ (Q ). Therefore taking Qλ = V −1 , we get Qλ P1−1 , P2−1 . Also, we have

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σ(P1 ) ⊆ σ(P2 ), since P2 = Q1 P1 U . Compute σ(P2 ). We have P2 = U V = Q1 P1 U , V −1 ∈ T (Q ), hence σ(P2 ) = σ(V )∪τ (U ∗V ) ⊆ σ(Qλ )∪τ (U ∗V ) by Lemma 2.6(a) and Lemma 2.7. (QP1 P2 ) = σ(Q) ∪ σ(P2 ) ∪ τ (Q ∗ P1 ) ∪ τ(P1 ∗ Consequently, σ P2 )∪τ (P2 ∗Q) = σ(Q)∪σ(Qλ )∪ τ (U ∗V )∪τ (Q∗P1 )∪τ (P1 ∗P2 )∪τ (P2 ∗Q) . But τ (U ∗ V ) ∈ σ(P2 ) and τ (Q ∗ P1 ) = τ (Q1 ∗ P1 ) ∈ σ(P2 ). Also, τ (P1 ∗ P2 ) = τ (P1 ∗ U ) by Lemma 2.6(b), as U ∈ H(P2 ), hence τ (P1 ∗ P2 ) ∈ σ(P2 ). Therefore, we get σ (QP1 P2 ) = σ(Q) ∪ σ(Qλ ) ∪ τ (P2 ∗ Q). Thus, |σ(Q) ∪ σ(Qλ )| ≥ | σ (R)| − 1 ≥ | σ (R)| − 3. / P2 and v2 ∈ P1 . Then Main Case 3. v1 ∈ P1 = XY, P1 = Y P2 Q0 , Q0 ∈ H(Q), P2 = Q1 U, P1 = U V, X −1 ∈ H(Qr ).

(3.21)

Since P1 and P1 overlap, ε(P1 ) = 1 then the ﬁrst two equations of (3.21) yield X ∼ P2 Q0 ∼ P1 , hence taking Qρ = X −1 , gives Qρ P1 , P2 . Now, it follows from (3.21) that σ(P2 ) ⊆ σ(P1 ) and σ(P1 ) = σ(X) ∪ τ (X ∗ Y ). Also, τ (Q ∗ P1 ) = τ (Q1 ∗U ) ∈ σ(P2 ), since Q1 ∈ τ (Q) and U ∈ H(P1 ), τ (P1 ∗P2 ) = τ (Y ∗P2 ) ∈ σ(P1 ) since Y ∈ T (P1 ) by (3.21) and ﬁnally τ (P2 ∗Q) = τ (P2 ∗Q0) ∈ σ(P1 ). Consequently, (QP1 P2 ) ⊆ σ(Q)∪σ(Qr )∪τ (X ∗ Y ), σ (QP1 P2 ) = σ(Q)∪σ(X)∪τ (X ∗ Y ). Hence, σ since X −1 ∈ H(Qr ), by (3.21). But then |σ(Q) ∪ σ(Qr )| ≥ | σ (QP1 P2 )| − 1, by Lemma 2.6(c), and the result follows. Case (2, 4). P2 is a subword of P1 . Assume ﬁrst that v2 is not an inner vertex of P1 . Then P1 = XY, P1 = Y Z, P2 = ZW and P1 = U P2 V.

(3.22)

ε(P1 )

Since = 1 then the ﬁrst two equations of (3.22) imply P1 ∼ X ∼ Z Y and the last equation of (3.22) implies P1 P2 . Consequently, X P1 , P2 . Deﬁne Qρ = X −1. Then Qρ P1 , P2 . Assume now that v2 is an inner vertex of P1 . Since P1 and P1 overlap, ε1 = 1. Hence, P1 = XY, P1 = Y P2 Q0 , Q0 = 1, Q0 ∈ H(Q) and P1 = U P2 V.

(3.23)

The ﬁrst two equations of (3.23) imply X = KL, P2 Q0 = LK and Y = (KL)α K, α ≥ 0. Consequently, Q0 P2 , P1 , L, K, X, Y , hence Q Q0 P1 , P2 . Now it follows from (3.22) that σ(P1 ) = σ(X) ∪ τ (X ∗ Y ) and σ(P2 ) ⊆ σ(P1 ). Also, it follows from (3.23) and Lemma 2.6(b) that τ (P1 ∗ P2 ) = τ (Y ∗ Z) ∈σ(P1 ) and τ (P2 ∗ Q) = τ (P2 ∗ Q0 ) ∈ σ(P1 ). Therefore, σ (QP1 P2 ) = σ(Q) ∪ σ(P1 ) ∪ τ (X ∗ Y ) ∪ τ (Q ∗ P1 ) ⊆ σ(Q) ∪ σ(Qr ) ∪ τ (X ∗ Y ) ∪ τ (Q ∗ P1 ). Hence |σ(Q) ∪ σ(Qr )| ≥ | σ (R)| − 2, by Lemma 2.6(c) and the result follows. Case (2, 5). / P1 . Assume v2 ∈ Since P1 and P1 overlap, ε(P1 ) = 1. Hence P1 = XY, P1 = Y Z and P2 = ZW .

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By the ﬁrst two equations, via Lemma 2.3(a) P1 ∼ Z and by the third equation P2 Z. Hence P1 ∼ Z ≺ P2 ≺ Q. Consequently, Q P2 P1 , hence P1 , P2 ≺ Q. Since P2 is a subword of Q, by Lemma 2.6(a), we have σ(P2 ) ⊆ σ(Q). In particular, σ(Z) ⊆ σ(Q). Now, σ(P1 ) = σ(Z) ∪ τ (Y ∗ Z) by Lemma 2.7 and τ (Y ∗ Z) = τ (P1 ∗ P2 ), by Lemma 2.6(b), since Y ∈ T (P1 ) and Z ∈ H(P2 ). Hence σ(P1 ) ⊆ σ(Q) ∪ τ (P1 ∗ P2 ). Consequently, σ (QP1 P2 ) = σ(Q) ∪ τ (P1 ∗ P2 ) ∪ τ (Q ∗ P1 ) ∪ τ (P2 ∗ Q), hence by Lemma 2.6(c) |σ(Q)| ≥ | σ (R)| − 3, as required. Assume now that v2 ∈ P1 . Then P1 = XY, P1 = Y P2 Q0 , Q0 = 1, Q0 ∈ H(Q). Hence P2 Q0 P1 , by Lemma 2.2(a). If ε(P2 ) = 1 then Q P2 , hence Q P2 Q0 P1 , i.e., Q P1 , P2 . Suppose ε(P2 ) = −1. By Lemma 2.1 X = KL, Y = (KL)α K, α ≥ 0 and P2 Q0 = LK. Consequently, P2 Q0 P1 P2 hence Q P1 , P2 . Now, by Lemma 2.7 σ(P1 ) =

σ(P2 Q0 ) ∪ τ (Y ∗ P2 )

= σ(P2 ) ∪ σ(Q0 ) ∪ τ (P2 ∗ Q) ⊆ σ(Q) ∪ τ (P2 ∗ Q), since σ(P2 ) ⊆ σ(Q), by Lemma 2.6(a). Therefore, σ (P1 P2 Q) = σ(Q)∪ τ (P1 ∗P2 ∗Q), hence |σ(Q)| ≥ | σ (R)| − 3, by Lemma 2.6(c). Case (3, 3 ). The cases when v1 ∈ P1 or v1 ∈ P2 can be dealt with in a way similar to Subcases 2 / P1 and v1 ∈ / P2 . We have and 3 of Case (2, 2 ). So we concentrate on the case v1 ∈ P1 = XU, P1 = Y Q0 , P2 = V Y, P2 = Q1 X where Q0 ∈ H(Q) and Q1 ∈ T (Q). Due to the second and fourth equations, it is enough to prove X, Y ≺ Q. Now, σ(P1 ) = σ(Y ) ∪ σ(Q0 ) ∪ τ (P2 ∗ Q) and σ(P2 ) = σ(X) ∪ σ(Q1 ) ∪ τ (Q ∗ P1 ). Hence σ (QP1 P2 ) = σ(X) ∪ σ(Y ) ∪ σ(Q) ∪ τ(P1 ∗ P2 ∗ Q).

(3.24)

Subcase 1. ε1 = ε2 = 1. Then Q1 X = V Y and XU = Y Q0 . If X = Y then Q1 = V and U = Q0 , hence h(U ) = h(Q0 ) = h(Q), violating the right normalization of the piece pair (P2 , P2 ) (see Deﬁnition 2.9). If X = Y Z, Z = 1, then Q0 = ZU, Q1 Y Z = V Y , hence V = Q1 T, Y Z = T Y and, by Lemma 2.3, Y ≺ Z ∼ T . Therefore X = Y Z implies Y ≺ Z ∼ X ∼ T . But Q0 = ZU implies Q0 Z. Consequently, Q Q0 X, Y . Also, σ(Z) ⊆ σ(Q0 ) and X = Y Z = T Y imply σ(X) = σ(Z)∪τ (Y ∗Z) ⊆ σ(Q0 )∪τ (Y ∗Z) ⊆ σ(Q) ∪ τ (P2 ∗ Q), by Lemma 2.7. Since σ(Y ) ⊆ σ(X) it follows from (3.24) that σ (QP1 P2 ) = σ(Q) ∪ τ(P1 ∗ P2 ∗ Q), hence |σ(Q)| ≥ | σ (R)| − 3. If Y = XZ, Z = 1, then U = ZQ0 , Q1 X = V XZ. Hence Q1 = V T, T X = XZ and, as in the previous case, T ∼ X ∼ Z ≺ Q1 , hence Y = XZ ≺ Q1 , by Lemma 2.2. Thus, X, Y ≺ Q. Since Y = T X = XZ, σ(Y ) = σ(XZ) = σ(T ) ∪ τ (T ∗ X) = σ(T ) ∪ τ (Q ∗ P1 ), i.e., σ(Y ) = σ(T ) ∪ τ (Q ∗ P1 ). But Q1 = V T implies σ(T ) ⊆ σ(Q1 ) ⊆ σ(Q), hence since σ(X) ⊆ σ(Y ) by (3.24) σ (R) = σ(X) ∪ σ(Y ) ∪ τ(QP1 P2 ) = σ(Q) ∪ τ(Q ∗ P1 ∗ P2 ) and again the result follows, by Lemma 2.6(c). Subcase 2. ε1 = 1 and ε2 = −1. Then XU = Y Q0 and V Y = X −1Q−1 1 . Suppose ﬁrst X = Y . Then U = Q0 and

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V X = X −1 Q1 . If V = X −1V1 then Q1 = V1 X, hence Q1 X, Y . If X −1 = V Z, then X = ZQ1 , hence X = ZQ1 = Z −1 V −1 . Since R contains no elements of order two, hence, this case can not occur. Hence V = X −1 V1 , V1 X = Q−1 . Consequently, σ(X) ⊆ σ(Q) and the result follows by (3.24), as above. If X = Y Z, −1 Z ∈ H(Q0 ) then V Y = Z −1 Y −1 Q−1 1 , hence Y ∈ T (Q1 ) and hence σ(Y ) ⊆ σ(Q) and σ(X) = σ(Y Z) ⊆ σ(Y ) ∪ σ(Q) ∪ τ (Y ∗ Z) = σ(Q) ∪ τ (P2 ∗ Q). Hence the result follows from (3.24), as above. Finally, if Y = XU0 , U = U0 Q0 then −1 V Y = V XU0 = X −1 Q−1 V1 , Q−1 = V1 XU0 = V1 Y . In 1 . Therefore V = X 1 particular σ(Y ), σ(X) ⊆ σ(Q1 ) ⊆ σ(Q) and the result follows by (3.24). Subcase 3. ε1 = −1 and ε2 = 1. This subcase dual to Subcase 2. Subcase 4. ε1 = −1 and ε2 = −1. This subcase follows directly from the four equations at the beginning of Case (3, 3 ). Case (3, 4). We have P2 = U V, P1 = V Q0 , P1 = HP2 T , hence σ(P2 ) ⊆ σ(P1 ). There are four cases to check, according as ε1 = ±1 and ε2 = ±1. We shall check only the cases ε1 = 1 and ε2 = ±1. The other two cases are similar. Subcase 1. ε1 = 1, ε2 = 1. Then H(U V )T = V Q0 . Hence Q0 = Q0 T and V Q0 = HU V . Consequently, Q0 V, H, U , hence Q P1 , P2 . Also, V Q0 = HU V implies by Lemma 2.7 (with A = HU , B = V , C = Q0 ) σ(V Q0 ) = σ(Q0 ) ∪ τ (V ∗ Q0 ). Since τ (V ∗ Q0 ) = τ (P2 ∗ Q), we get σ(V ) ⊆ σ(Q) ∪ τ (P2 ∗ Q). Now σ (R) = σ(P1 ) ∪ σ(P2 ) ∪ σ(Q) ∪ τ(Q ∗ P1 ∗ P2 ) = σ(P1 ) ∪ σ(Q) ∪ τ(Q ∗ P1 ∗ P2 ). But σ(P1 ) = σ(P1 ) = σ(V Q0 ) = σ(V )∪σ(Q0 )∪τ (V ∗Q0 ) ⊆ σ(V )∪σ(Q)∪τ (P2 ∗Q) ⊆ σ(Q)∪τ (P2 ∗Q). Consequently, σ (R) = σ(Q) ∪ τ(Q ∗ P1 ∗ P2 ) and the result follows by Lemma 2.6(c) as in previous cases. Subcase 2. ε1 = 1, ε2 = −1. Then H(V −1 U −1 )T = V Q0 , hence Q0 = Q0 U −1 T and V Q0 = HV −1 . Since F has no elements of order two, H = V H1 and Q0 = H1 V −1 . Consequently, Q V, H1 , hence Q U, V, P1 , P2 . Also, σ(P1 ) = σ(V ) ∪ σ(Q0 ) ∪ τ (V ∗ Q0 ) ⊆ σ(Q) ∪ τ (P2 ∗ Q). Therefore, σ (QP1 P2 ) = σ(Q) ∪ τ(Q ∗ P1 ∗ P2 ) and the result follows, by Lemma 2.6(c). Case (3, 5 ). Then P1 = V Q0 , Q0 ∈ H(Q), P2 = U V , hence because P2 is a subword of Q, Q P2 . Consequently, Q U, V , hence Q P1 , P2 . Also, σ(P2 ) ⊆ σ(Q) and σ(P1 ) = σ(V ) ∪ σ(Q0 ) ∪ τ (V ∗ Q0 ) ⊆ σ(Q) ∪ τ (P2 ∗ Q). Consequently, σ (QP1 P2 ) = σ(Q) ∪ τ(Q ∗ P1 ∗ P2 ) and the result follows, by Lemma 2.6(c). Case (4, 4 ). Then P1 is a subword of P2 and P2 is a subword of P1 . Consequently, |P1 | = |P2 | and since R is reduced ε1 = ε2 = 1. Therefore, P1 = P2 and if v0 has valency

202

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three in H then Qr has P1−1 as a head. Therefore, σ(P1 ) = σ(P2 ) ⊆ σ(Qr ) and as ⊆ σ(Q) ∪ σ(Qr ) ∪ τ(P1 ∗ P2 ∗ Q). Similarly, if v2 has valency three in result σ(R) ⊆ σ(Q) ∪ σ(Q ) ∪ τ(P1 ∗ P2 ∗ Q). In both cases the result follows, by H then σ(R) Lemma 2.6(c). Cases (4, 5 ) and (5, 5 ). = σ(Q) ∪ τ(Q ∗ P1 ∗ P2 ). It follows immediately that Q P1 , P2 and σ(R) The proposition is proved.

Acknowledgement I am grateful to the referee for his useful comments regarding the presentation of the work.

References [J1] Juh´ asz, A. Small cancellation theory with uniﬁed small cancellation condition. J. London. Math. Soc. 2 (40):57–80 (1989). [J2] Juh´ asz, A. Solution of the membership problem of Magnus subgroups of onerelator free products with small cancellation condition. In: Algebra and Geometry in Geneva and Barcelona, “Asymptotic and Probabilistic Methods in Geometric Group Theory”, Geneva, June 2005 and “Barcelona Group Theory Conference”, Barcelona, July 2005, 169–195, Birkh¨ auser (2007). [J3] Juh´ asz, A. A Freiheitssatz for Whitehead graphs of one-relator groups with small cancellation. Communications in Algebra 37:8, 2714–2741 (2009). [Jo] Johnson, D.L. Topics in the Theory of Group Presentations. LMSLNS, Cambridge University Press, Cambridge. 42 (1980). [L-S] Lyndon, R.C., Schupp, P.E. Combinatorial Group Theory, Springer-Verlag, BerlinHeidelberg-New York, (1977). [M] Magnus, W., Das Identit¨ atsproblem f¨ ur Gruppen mit einer deﬁnierenden Relation, Math. Ann. 106 pp. 295–307 (1932). Arye Juh´ asz Department of Mathematics Technion – Israel Institute of Technology 32000 Haifa, Israel e-mail: [email protected]

Combinatorial and Geometric Group Theory Trends in Mathematics, 203–242 c 2010 Springer Basel AG

Equations and Fully Residually Free Groups Olga Kharlampovich and Alexei G. Myasnikov Abstract. This paper represents notes of the mini-courses given by the authors at the GCGTA conference in Dortmund (2007), Ottawa-Saint Sauveur conference (2007), Escola d’Algebra in Rio de Janeiro (2008) and Alagna (Italy, 2008) conference on equations in groups. We explain here the Elimination process for solving equations in a free group which has Makanin-Razborov process as a prototype. We also explain how we use this process to obtain the structure theorem for ﬁnitely generated fully residually free groups and many other results. Mathematics Subject Classiﬁcation (2000). 20-02. Keywords. Equations, free groups.

1. Introduction 1.1. Motivation Solving equations is one of the main themes in mathematics. A large part of the combinatorial group theory revolves around the word and conjugacy problems – particular types of equations in groups. Whether a given equation has a solution in a given group is, as a rule, a non-trivial problem. A more general and more diﬃcult problem is to decide which formulas of the ﬁrst-order logic hold in a given group. Around 1945 A. Tarski put forward two problems on elementary theories of free groups that served as a motivation for much of the research in group theory and logic for the last sixty years. A joint eﬀort of mathematicians of several generations culminated in the following theorems, solving these Tarski’s conjectures. Theorem 1 (Kharlampovich and Myasnikov [44], Sela [60]). The elementary theories of non-Abelian free groups coincide. Theorem 2 (Kharlampovich and Myasnikov [44]). The elementary theory of a free group F (with constants for elements from F in the language) is decidable. Mini-course for the GCGTA conference in Dortmund (2007)and Escola d’Algebra in Rio de Janeiro (2008).

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We recall that the elementary theory T h(G) of a group G is the set of all ﬁrstorder sentences in the language of group theory which are true in G. A discussion of these conjectures can be found in several textbooks on logic, model and group theory (see, for example, [14], [22], [29]). The work on the Tarski conjectures was rather fruitful – several areas of group theory were developed along the way. It was clear from the beginning that to deal with the Tarski’s conjectures one needs to have at least two principal things done: a precise description of solution sets of systems of equations over free groups and a robust theory of ﬁnitely generated groups which satisfy the same universal (existential) formulas as a free non-Abelian group. In the classical case, algebraic geometry provides a unifying view-point on polynomial equations, while commutative algebra and the elimination theory give the required decision tools. Around 1998 three papers appeared almost simultaneously that address analogous issues in the group case. Basics of algebraic (or Diophantine) geometry over groups has been outlined by Baumslag, Myasnikov and Remeslennikov in [5], while the fundamentals of the elimination theory and the theory of fully residually free groups appeared in the works by Kharlampovich and Myasnikov [37], [38]. These two papers contain results that became fundamental for the proof of the above two theorems, as well as in the theory of fully residually free groups. The goal of these lectures is to explain why these results are important and to give some ideas of the proof. 1.2. Milestones of the theory of equations in free groups The ﬁrst general results on equations in groups appeared in the 1960’s [31]. About this time Lyndon (a former student of Tarski) came up with several extremely important ideas. One of these is to consider completions of a given group G by extending exponents into various rings (analogs of extension of ring of scalars in commutative algebra) and use these completions to parameterize solutions of equations in G. Another idea is to consider groups with free length functions with values in some ordered Abelian group. This allows one to axiomatize the classical Nielsen technique based on the standard length function in free groups and apply it to “non-standard” extensions of free groups, for instance, to ultrapowers of free groups. A link with the Tarski’s problems comes here by the Keisler-Shelah theorem, that states that two groups are elementarily equivalent if and only if their ultrapowers (with respect to a non-principal ultraﬁlter) are isomorphic. The idea to study freely discriminated (fully residually free) groups in connection to equations in a free group also belongs to Lyndon. He proved [30] that the completion F Z[t] of a free group F by the polynomial ring Z[t] (now it is called the Lyndon’s completion of F ) is discriminated by F . At the time the Tarski’s problems withstood the attack, but these ideas gave birth to several inﬂuential theories in modern algebra, which were instrumental in the recent solution of the problems. One of the main ingredients that was lacking at the time was a robust mechanism to solve equations in free groups and a suitable description of the solution sets of equations. The main

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technical goal of these lectures is to describe a host of methods that altogether give this mechanism, that we refer to as Elimination Processes. Also in 1960’s Malcev [34] described solutions of the equation zxyx−1 y −1 z −1 = aba−1 b−1 in a free group, which is the simplest non-trivial quadratic equation in groups. The description uses the group of automorphisms of the coordinate group of the equation and the minimal solutions relative to these automorphisms – a very powerful idea, that nowadays is inseparable from the modern approach to equations. The ﬁrst break-through on Tarski’s problem came from Merzljakov (who was a part of Malcev’s school in Novosibirsk). He proved [50] a remarkable theorem that any two non-Abelian free groups of ﬁnite rank have the same positive theory and showed that positive formulas in free groups have deﬁnable Skolem functions, thus giving quantiﬁer elimination of positive formulas in free groups to existential formulas. Recall that the positive theory of a group consists of all positive (without negations in their normal forms) sentences that are true in this group. These results were precursors of the current approach to the elementary theory of a free group. In the eighties new crucial concepts were introduced. Makanin proved [48] the algorithmic decidability of the Diophantine problem over free groups, and showed that both, the universal theory and the positive theory of a free group are algorithmically decidable. He created an extremely powerful technique (the Makanin elimination process) to deal with equations over free groups. Shortly afterwards, Razborov (at the time a PhD student of Steklov’s Institute, where Makanin held a position) described the solution set of an arbitrary system of equations over a free group in terms of what is known now as MakaninRazborov diagrams [54], [55]. A few years later Edmunds and Commerford [18] and Grigorchuck and Kurchanov [27] described solution sets of arbitrary quadratic equations over free groups. These equations came to group theory from topology and their role in group theory was not altogether clear then. Now they form one of the corner-stones of the theory of equations in groups due to their relations to JSJ-decompositions of groups. 1.3. New age These are milestones of the theory of equations in free groups up to 1998. The last missing principal component in the theory of equations in groups was a general geometric point of view similar to the classical aﬃne algebraic geometry. Back to 1970’s Lyndon (again!) was musing on this subject [32] but for no avail. Finally, in the late 1990’s Baumslag, Kharlampovich, Myasnikov, and Remeslennikov developed the basics of the algebraic geometry over groups [5, 37, 38, 39, 36], introducing analogs of the standard algebraic geometry notions such as algebraic sets, the Zariski topology, Noetherian domains, irreducible varieties, radicals and coordinate groups, rational equivalence, etc.

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With all this machinery in place it became possible to make the next crucial step and tie the algebraic geometry over groups, Makanin-Razborov process for solving equations, and Lyndon’s free Z[t]-exponential group F Z[t] into one closely related theory. The corner stone of this theory is Decomposition Theorem from [38] (see Section 4.2 below) which describes the solution sets of systems of equations in free groups in terms of non-degenerate triangular quasi-quadratic (NTQ) systems. The coordinate groups of the NTQ systems (later became also known as residually free towers) play a fundamental role in the theory of fully residually free groups, as well as in the elementary theory of free groups. The Decomposition Theorem allows one to look at the processes of the Makanin-Razborov’s type as non-commutative analogs of the classical elimination processes from algebraic geometry. With this in mind we refer to such processes in all their variations as Elimination Processes (EP). In the rest of the notes we discuss more developments of the theory, focusing mostly on the elimination processes, fully residually free (limit) groups, and new techniques that appear here.

2. Basic notions of algebraic geometry over groups Following [5] and [39] we introduce here some basic notions of algebraic geometry over groups. Let G be a group generated by a ﬁnite set A, F (X) be a free group with basis X = {x1 , x2 , . . . , xn }, we deﬁned G[X] = G ∗ F (X) to be a free product of G and F (X). If S ⊂ G[X] then the expression S = 1 is called a system of equations over G. As an element of the free product, the left side of every equation in S = 1 can be written as a product of some elements from X ∪ X −1 (which are called variables) and some elements from A (constants). To emphasize this we sometimes write S(X, A) = 1. A solution of the system S(X) = 1 over a group G is a tuple of elements g1 , . . . , gn ∈ G such that after replacement of each xi by gi the left-hand side of every equation in S = 1 turns into the trivial element of G. To study equations over a given ﬁxed group G it is convenient to consider the category of G-groups, i.e., groups which contain the group G as a distinguished subgroup. If H and K are G-groups then a homomorphism φ : H → K is a G-homomorphism if gφ = g for every g ∈ G, in this event we write φ : H →G K. In this category morphisms are G-homomorphisms; subgroups are G-subgroups, etc. A solution of the system S = 1 over G can be described as a G-homomorphism φ : G[X] −→ G such that φ(S) = 1. Denote by ncl(S) the normal closure of S in G[X], and by GS the quotient group G[X]/ncl(S). Then every solution of S(X) = 1 in G gives rise to a G-homomorphism GS → G, and vice versa. By VG (S) we denote the set of all solutions in G of the system S = 1, it is called the algebraic set deﬁned by S. This algebraic set VG (S) uniquely corresponds to the normal subgroup R(S) = {T (x) ∈ G[X] | ∀A ∈ Gn (S(A) = 1 → T (A) = 1}

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of the group G[X]. Notice that if VG (S) = ∅, then R(S) = G[X]. The subgroup R(S) contains S, and it is called the radical of S. The quotient group GR(S) = G[X]/R(S) is the coordinate group of the algebraic set V (S). Again, every solution of S(X) = 1 in G can be described as a G-homomorphism GR(S) → G. By HomG (H, K) we denote the set of all G-homomorphisms from H into K. It is not hard to see that the free product G ∗ F (X) is a free object in the category of G-groups. This group is called a free G-group with basis X, and we denote it by G[X]. A G-group H is termed ﬁnitely generated G-group if there exists a ﬁnite subset A ⊂ H such that the set G ∪ A generates H. We refer to [5] for a general discussion on G-groups. A group G is called a CSA group if every maximal Abelian subgroup M of G is malnormal, i.e., M g ∩ M = 1 for any g ∈ G − M. The abbreviation CSA means conjugacy separability for maximal Abelian subgroups. The class of CSA-groups is quite substantial. It includes all Abelian groups, all torsion-free hyperbolic groups, all groups acting freely on Λ-trees and many one-relator groups (see, for example, [26]. We deﬁne a Zariski topology on Gn by taking algebraic sets in Gn as a subbasis for the closed sets of this topology. Namely, the set of all closed sets in the Zariski topology on Gn can be obtained from the set of algebraic sets in two steps: 1) take all ﬁnite unions of algebraic sets; 2) take all possible intersections of the sets obtained in step 1). If G is a non-Abelian CSA group and we in the category of G-groups, then the union of two algebraic sets is again algebraic. Indeed, if {wi = 1, i ∈ I} and {uj = 1, j ∈ J} are systems of equations, then in a CSA group their disjunction is equivalent to a system [wi , uj ] = [wi , uaj ] = [wi , ubj ] = 1, i ∈ I, j ∈ J for any two non-commuting elements a, b from G. Therefore the closed sets in the Zariski topology on Gn are precisely the algebraic sets. A group G is called equationally Noetherian if every system S(X) = 1 with coeﬃcients from G is equivalent over G to a ﬁnite subsystem S0 = 1, where S0 ⊂ S, i.e., VG (S) = VG (S0 ). It is known that linear groups (in particular, freely discriminated groups) are equationally Noetherian (see [25], [11], [5]). If G is equationally Noetherian then the Zariski topology on Gn is Noetherian for every n, i.e., every proper descending chain of closed sets in Gn is ﬁnite. This implies that every algebraic set V in Gn is a ﬁnite union of irreducible subsets (they are called irreducible components of V ), and such decomposition of V is unique. Recall that a closed subset V is irreducible if it is not a union of two proper closed (in the induced topology) subsets.

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3. Fully residually free groups 3.1. Deﬁnitions and elementary properties Finitely generated fully residually free groups (limit groups) play a crucial role in the theory of equations and ﬁrst-order formulas over a free group. It is remarkable that these groups, which have been widely studied before, turn out to be the basic objects in newly developing areas of algebraic geometry and model theory of free groups. Recall that a group G is called fully residually free (or freely discriminated, or ω-residually free) if for any ﬁnitely many non-trivial elements g1 , . . . , gn ∈ G there exists a homomorphism φ of G into a free group F , such that φ(gi ) = 1 for i = 1, . . . , n. The next proposition summarizes some simple properties of fully residually free groups. Proposition 1. Let G be a fully residually free group. Then G possesses the following properties. 1) 2) 3) 4)

5) 6) 7) 8) 9)

G is torsion-free; Each subgroup of G is a fully residually free group; G has the CSA property; Each Abelian subgroup of G is contained in a unique maximal ﬁnitely generated Abelian subgroup, in particular, each Abelian subgroup of G is ﬁnitely generated; G is ﬁnitely presented, and has only ﬁnitely many conjugacy classes of its maximal Abelian subgroups. G has solvable word problem; G is linear; Every 2-generated subgroup of G is either free or Abelian; If rank(G) = 3 then either G is free of rank 3, free Abelian of rank 3, or a free rank one extension of centralizer of a free group of rank 2 (that is G = x, y, t|[u(x, y), t] = 1, where the word u is not a proper power).

Properties 1 and 2 follow immediately from the deﬁnition of an F -group. A proof of property 3 can be found in [5]; property 4 is proven in [38]. Properties 4 and 5 are proved in [38]. Solvability of the word problem follows from [49] or from residual ﬁniteness of a free group. Property 9 is proved in [23]. Property 7 follows from linearity of F and property 6 in the next proposition. The ultraproduct of ∗ ∗ SL 2 (Z) is SL2 ( Z), where Z is the ultpaproduct of Z. (Indeed, the direct product SL2 (Z) is isomorphic to SL2 ( Z). Therefore, one can deﬁne a homomorphism from the ultraproduct of SL2 (Z) onto SL2 (∗ Z). Since the intersection of a ﬁnite number of sets from an ultraﬁlter again belongs to the ultraﬁlter, this epimorphism is a monomorphism.) Being ﬁnitely generated G embeds in SL2 (R), where R is a ﬁnitely generated subring in ∗ Z. Proposition 2. (no coeﬃcients) Let G be a ﬁnitely generated group. Then the following conditions are equivalent:

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1) G is freely discriminated (that is for ﬁnitely many non-trivial elements g1 , . . . , gn ∈ G there exists a homomorphism φ from G to a free group such that φ(gi ) = 1 for i = 1, . . . , n); 2) [Remeslennikov] G is universally equivalent to F (in the language without constants); 3) [Baumslag, Kharlampovich, Myasnikov, Remeslennikov] G is the coordinate group of an irreducible variety over a free group. 4) [Sela] G is a limit group (to be deﬁned in the proof of Proposition 3). 5) [Champetier and Guirardel] G is a limit of free groups in Gromov-Hausdorﬀ metric (to be deﬁned in the proof of Proposition 3). 6) G embeds into an ultrapower of free groups. Proposition 3. (with coeﬃcients) Let G be a ﬁnitely generated group containing a free non-Abelian group F as a subgroup. Then the following conditions are equivalent: 1) G is F -discriminated by F ; 2) [Remeslennikov] G is universally equivalent to F (in the language with constants); 3) [Baumslag, Kharlampovich, Myasnikov, Remeslennikov] G is the coordinate group of an irreducible variety over F . 4) [Sela] G is a restricted limit group. 5) [Champetier and Guirardel] G is a limit of free groups in Gromov-Hausdorﬀ metric. 6) G F -embeds into an ultrapower of F . We will prove Proposition 3, the proof of Proposition 2 is very similar. We will ﬁrst prove the equivalence 1)⇔ 2). Let L A be the language of group theory with generators A of F as constants. Let G be a f.g. group which is F -discriminated by F . Consider a formula ∃X(U (X, A) = 1 ∧ W (X, A) = 1). If this formula is true in F , then it is also true in G, because F ≤ G. If it is ¯ ∈ Gm holds U (X, ¯ A) = 1 and W (X, ¯ A) = 1. Since true in G, then for some X G is F -discriminated by F , there is an F -homomorphism φ : G → F such that ¯ A)) = 1, i.e., W (X ¯ φ , A) = 1. Of course U (X ¯ φ , A) = 1. Therefore the φ(W (X, above formula is true in F . Since in F -group a conjunction of equations [inequalities] is equivalent to one equation [resp., inequality], the same existential sentences in the language LA are true in G and in F . Suppose now that G is F -universally equivalent to F . Let G = X, A | S(X, A) = 1, be a presentation of G and w1 (X, A), . . . , wk (X, A) nontrivial elements in G. Let Y be the set of the same cardinality as X. Consider a system of equations S(Y, A) = 1 in variables Y in F . Since the group F is equationally Noetherian, this system is equivalent over F to a ﬁnite subsystem S1 (Y, A) = 1.

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The formula

Ψ = ∀Y (S1 (Y, A) = 1 → w1 (Y, A) = 1 ∨ · · · ∨ wk (Y, A) = 1) .

is false in G, therefore it is false in F . This means that there exists a set of elements B in F such that S1 (B, A) = 1 and, therefore, S(B, A) = 1 such that w1 (B, A) = 1 ∧ · · · ∧ wk (B, A) = 1. The map X → B that is identical on F can be extended to the F -homomorphism from G to F . 1)⇔ 3) Let H be an equationally Noetherian CSA-group. We will prove that V (S) is irreducible if and only if HR(S) is discriminated in H by H-homomorphisms. Suppose V (S) is not irreducible and V (S) = V (S1 ) ∪ V (S2 ) is its decomposition into proper subvarieties. Then there exist si ∈ R(Si ) \ R(Sj ), j = i. The set {s1 , s2 } cannot be discriminated in H by H-homomorphisms. Suppose now s1 , . . . , sn are elements such that for any retract f : HR(S) → H m there exists i such that f (si ) = 1; then V (S) = i=1 V (S ∪ si ). Sela [57] deﬁned limit groups as follows. Let G be a f.g. group and let {φj } be a sequence of homomorphisms from G to a free group F belonging to distinct conjugacy classes (distinct F -homomorphisms belong to distinct conjugacy classes). F acts by isometries on its Cayley graph X which is a simplicial tree. Hence, there is a sequence of actions of G on X corresponding to {φj }. By rescaling metric on X for each φj one obtains a sequence of simplicial trees {Xj } and a corresponding sequence of actions of G. {Xj } converges to a real tree Y (Gromov-Hausdorﬀ limit) endowed with an isometric action of G. The kernel of the action of G on Y is deﬁned as K = {g ∈ G | gy = y, ∀y ∈ Y }. Finally, G/K is said to be the limit group (corresponding to {φj } and rescaling constants). We will prove now the equivalence 1)⇔ 4). A slight modiﬁcation of the proof below should be made to show that limit groups are exactly f.g. fully residually free groups. Suppose that G = g1 , . . . , gk is f.g. and discriminated by F . There exists a sequence of homomorphisms φn : G → F, so that φn maps the elements in a ball of radius n in the Cayley graph of G to distinct elements in F . By rescaling the metric on F , we obtain a subsequence of homomorphisms φm which converges to an action of a limit group L on a real tree Y . In general, L is a quotient of G, but since the homomorphisms were chosen so that φn maps a ball of radius n monomorphically into F , G is isomorphic to L and, therefore, G is a limit group. To prove the converse, we need the fact (ﬁrst proved in [57]) that a f.g. limit group is ﬁnitely presented. We may assume further that a limit group G is nonAbelian because the statement is, obviously, true for Abelian groups. By deﬁnition, there exists a f.g. group H, an integer k and a sequence of homomorphisms hk : H → F , so that the limit of the actions of H on the Cayley graph of F via the homomorphisms hk is a faithful action of G on some real tree Y . Since G is ﬁnitely

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presented for all but ﬁnitely many n, the homomorphism hn splits through the limit group G, i.e., hn = φψn , where φ : H → G is the canonical projection map, and the ψn ’s are homomorphisms ψn : G → F . If g = 1 in G, then for all but ﬁnitely many ψn ’s g ψn = 1. Hence, for every ﬁnite set of elements g1 , . . . , gm = 1 in G for ψn = 1, so G is F discriminated. all but ﬁnitely many indices n, g1ψn , . . . , gm The equivalence 2)⇔ 6) is a particular case of general results in model theory (see for example [4] Lemma 3.8 Chap.9). 5)⇔ 6). Champetier and Guirardel [16] used another approach to limit groups. A marked group (G, S) is a group G with a prescribed family of generators S = (s1 , . . . , sn ). Two marked groups (G, (s1 , . . . , sn )) and (G , (s1 , . . . , sn )) are isomorphic as marked groups if the bijection si ←→ si extends to an isomorphism. For example, ( a, (1, a)) and ( a, (a, 1)) are not isomorphic as marked groups. Denote by Gn the set of groups marked by n elements up to isomorphism of marked groups. One can deﬁne a metric on Gn by setting the distance between two marked groups (G, S) and (G , S ) to be e−N if they have exactly the same relations of length at most N (under the bijection S ←→ S ). Finally, a limit group in their terminology is a limit (with respect to the metric above) of marked free groups in Gn . It is shown in [16] that a group is a limit group if and only if it is a ﬁnitely generated subgroup of an ultraproduct of free groups (for a non-principal ultraﬁlter), and any such ultraproduct of free groups contains all the limit groups. This implies the equivalence 5)⇔ 6). Notice that ultrapowers of a free group have the same elementary theory as a free group by Los’ theorem. First non-free ﬁnitely generated examples of fully residually free groups, that include all non-exceptional surface groups, appeared in [2], [3]. They obtained fully residually free groups as subgroups of free extensions of centralizers in free groups. 3.2. Lyndon’s completion F Z[t] Studying equations in free groups Lyndon [33] introduced the notion of a group with parametric exponents in an associative unitary ring R. It can be deﬁned as a union of the chain of groups F = F0 < F1 < · · · < Fn < · · · , where F = F (X) is a free group on an alphabet X, and Fk is generated by Fk−1 and formal expressions of the type {wα | w ∈ Fk−1 , α ∈ R}. That is, every element of Fk can be viewed as a parametric word of the type αm , w1α1 w2α2 · · · wm

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where m ∈ N, wi ∈ Fk−1 , and αi ∈ R. In particular, he described free exponential groups F Z[t] over the ring of integer polynomials Z[t]. Notice that ultrapowers of free groups are operator groups over ultraproducts of Z. In the same paper Lyndon proved an amazing result that F Z[t] is fully residually free. Hence all subgroups of F Z[t] are fully residually free. Lyndon showed that solution sets of one variable equations can be described in terms of parametric words. Later it was shown in [1] that coordinate groups of irreducible one-variable equations are just extensions of centralizers in F of rank one (see the deﬁnition in the second paragraph below). In fact, this result is not entirely accidental, extensions of centralizers play an important part here. Recall that Baumslag [2] already used them in proving that surface groups are freely discriminated. Now, breaking the natural march of history, we go ahead of time and formulate one crucial result which justiﬁes our discussion on Lyndon’s completion F Z[t] and highlights the role of the group F Z[t] in the whole subject. Theorem (The Embedding Theorem [38],[39]) Given an irreducible system S = 1 over F one can eﬀectively embed the coordinate group FR(S) into F Z[t] . A modern treatment of exponential groups was done by Myasnikov and Remeslennikov [35] who proved that the group F Z[t] can be obtained from F by an inﬁnite chain of HNN-extensions of a very speciﬁc type, so-called extensions of centralizers: F = G0 < G1 < · · · < · · · ∪ Gi = F Z[t] where Gi+1 = Gi , ti | [CGi (ui ), ti ] = 1. (extension of the centralizer CGi (ui ), where ui ∈ Gi ). This implies that ﬁnitely generated subgroups of F Z[t] are, in fact, subgroups of Gi . Since Gi in an HNN-extension, one can apply Bass-Serre theory to describe the structure of these subgroups. In fact, f.g. subgroups of Gi are fundamental groups of graphs of groups induced by the HNN structure of Gi . For instance, it is routine now to show that all such subgroups H of Gi are ﬁnitely presented. Indeed, we only have to show that the intersections Gi−1 ∩ H g are ﬁnitely generated. Notice, that if in the amalgamated product amalgamated subgroups are ﬁnitely generated and one of the factors is not, then the amalgamated product is not ﬁnitely generated (this follows from normal forms of elements in the amalgamated products). Similarly, the base group of a f.g. HNN extension with f.g. associated subgroups must be f.g. Earlier Pfander [52] proved that f.g. subgroups of the free Z[t]-group on two generators are ﬁnitely presented. Description of f.g. subgroups of F Z[t] as fundamental groups of graphs of groups implies immediately that such groups have non-trivial Abelian splittings (as amalgamated product or HNN extension with Abelian edge group which is maximal Abelian in one of the base subgroups). Furthermore, these groups can be obtained from free groups by ﬁnitely many free constructions (see next section). The original Lyndon’s result on fully residual freeness of F Z[t] gives decidability of the Word Problem in F Z[t] , as well as in all its subgroups. Since any

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fully residually free group given by a ﬁnite presentation with relations S can be presented as the coordinate group FR(S) of a coeﬃcient-free system S = 1. The Embedding Theorem then implies decidability of WP in arbitrary f.g. fully residually residually free group. The Conjugacy Problem is also decidable in F Z[t] – but this was proved much later, by Ribes and Zalesski in [59]. A similar, but stronger, result is due to Lyutikova who showed in [47] that the Conjugacy Problem in F Z[t] is residually free, i.e., if two elements g, h are not conjugate in F Z[t] (or in Gi ) then there is an F -epimorphism φ : F Z[t] → F such that φ(g) and φ(h) are not conjugate in F . Unfortunately, this does not imply immediately that the CP in subgroups of F Z[t] is also residually free, since two elements may be not conjugated in a subgroup H ≤ F Z[t] , but conjugated in the whole group F Z[t] . We discuss CP in arbitrary f.g. fully res. free groups in Section 5.

4. Main results in [38] 4.1. Structure and embeddings In 1996 we proved the converse of the Lyndon’s result mentioned above, every ﬁnitely generated fully residually free group is embeddable into F Z[t] . Theorem 3 ([38], [39]). Given an irreducible system S = 1 over F one can eﬀectively embed the coordinate group FR(S) into F Z[t] , i.e., one can ﬁnd n and an embedding FR(S) → Gn into an iterated centralizer extension Gn . Corollary 1. For every freely indecomposable non-Abelian ﬁnitely generated fully residually free group one can eﬀectively ﬁnd a non-trivial splitting (as an amalgamated product or HNN extension) over a cyclic subgroup. Corollary 2. Every ﬁnitely generated fully residually free group is ﬁnitely presented. There is an algorithm that, given a presentation of a f.g. fully residually free group G and generators of the subgroup H, ﬁnds a ﬁnite presentation for H. Corollary 3. Every ﬁnitely generated residually free group G is a subgroup of a direct product of ﬁnitely many fully residually free groups; hence, G is embeddable into F Z[t] × · · · × F Z[t] . If G is given as a coordinate group of a ﬁnite system of equations, then this embedding can be found eﬀectively. Indeed, there exists a ﬁnite system of coeﬃcient free equations S = 1 such that G is a coordinate group of this system, and ncl(S) = R(S). If V (S) = ∪ni=1 V (Si ) is a representation of V (S) as a union of irreducible components, then R(S) = ∩ni=1 R(Si ) and G embeds into a direct product of coordinate groups of systems Si = 1, i = 1, . . . , n. This allows one to study the coordinate groups of irreducible systems of equations (fully residually free groups) via their splittings into graphs of groups. This also provides a complete description of ﬁnitely generated fully residually free groups and gives a lot of information about their algebraic structure. In particular,

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they act freely on Zn -trees, and all these groups, except for Abelian and surface groups, have a non-trivial cyclic JSJ-decomposition. Let K be an HNN-extension of a group G with associated subgroups A and B. K is called a separated HNN-extension if for any g ∈ G, Ag ∩ B = 1. Corollary 4. Let a group G be obtained from a free group F by ﬁnitely many centralizer extensions. Then every f.g. subgroup H of G can be obtained from free Abelian groups of ﬁnite rank by ﬁnitely many operations of the following type: free products, free products with Abelian amalgamated subgroups at least one of which is a maximal Abelian subgroup in its factor, free extensions of centralizers, separated HNN-extensions with Abelian associated subgroups at least one of which is maximal. Corollary 5 (Groves, Wilton [28]). One can enumerate all ﬁnite presentations of fully residually free groups. Theorem 3 is proved as a corollary of Theorem 6 below. Corollary 6. Every f.g. fully residually free group acts freely on some Zn -tree with lexicographic order for a suitable n. Hence, a simple application of the change of the group functor shows that H also acts freely on an Rn -tree. Recently, Guirardel proved the latter result independently using diﬀerent techniques [24]. It is worthwhile to mention here that free group actions on Zn -trees give a tremendous amount of information on the group and its subgroups, especially with regard to various algorithmic problems (see Section 5). Notice, that there are f.g. groups acting freely on Zn -trees which are not fully residually free (see conjecture (2) from Sela’s list of open problems). The simplest example is the group of closed non-orientable surface of genus 3. In fact, the results in [45, 46] show that there are very many groups like that – the class of groups acting freely on Zn -trees is much wider than the class of fully residually free groups. This class deserves a separate discussion, for which we refer to [45, 46]. Combining Corollary 4 with the results from [40] or [8] we proved in [38] that f.g. fully residually free groups without subgroups Z × Z (or equivalently, with cyclic maximal Abelian subgroups) are hyperbolic. We will see in Section 4.2 that this has some implication on the structure of the models of the ∀∃-theory of a given non-Abelian free group. Later, Dahmani [21] proved a generalization of this, namely, that an arbitrary f.g. fully residually free group is hyperbolic relative to its maximal Abelian non-cyclic subgroups. Recently N. Touikan described coordinate groups of two-variable equations [61]. 4.2. Triangular quasi-quadratic systems We use an Elimination Process to transform systems of equations. Elimination Process EP is a symbolic rewriting process of a certain type that transforms formal systems of equations in groups or semigroups. Makanin (1982) introduced the initial version of the EP. This gives a decision algorithm to verify consistency of a

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given system – decidability of the Diophantine problem over free groups. He estimates the length of the minimal solution (if it exists). Makanin introduced the fundamental notions: generalized equations, elementary and entire transformations, notion of complexity. Razborov (1987) developed EP much further. Razborov’s EP produces all solutions of a given system in F . He used special groups of automorphisms, and fundamental sequences to encode solutions. We obtained in 1996 [38] an eﬀective description of solutions of equations in free (and fully residually free ) groups in terms of very particular triangular systems of equations. First, we give a deﬁnition. Triangular quasi-quadratic (TQ) system is a ﬁnite system that has the following form S1 (X1 , X2 , . . . , Xn , A) = 1, S2 (X2 , . . . , Xn , A) = 1, ··· Sn (Xn , A) = 1 where either Si = 1 is quadratic in variables Xi , or Si = 1 is a system of commutativity equations for all variables from Xi and, in addition, equations [x, u] = 1 for all x ∈ Xi and some u ∈ FR(Si+1 ,...,Sn ) or Si is empty. A TQ system above is non-degenerate (NTQ) if for every i, Si (Xi , . . ., Xn , A) = 1 has a solution in the coordinate group FR(Si+1 ,...,Sn ) , i.e., Si = 1 (in algebraic geometry one would say that a solution exists in a generic point of the system Si+1 = 1, . . . , Sn = 1). We proved in [37] (see also [39]) that NTQ systems are irreducible and, therefore, their coordinate groups (NTQ groups) are fully residually free. (Later Sela called NTQ groups ω-residually free towers [57].) We represented a solution set of a system of equations canonically as a union of solutions of a ﬁnite family of NTQ groups. Theorem 4 ([38], [39]). One can eﬀectively construct EP that starts on an arbitrary system S(X, A) = 1 and results in ﬁnitely many NTQ systems U1 (Y1 ) = 1, . . . , Um (Ym ) = 1 such that VF (S) = P1 (V (U1 )) ∪ · · · ∪ Pm ((Um )) ¯ ∈ for some word mappings P1 , . . . , Pm . (Pi maps a tuple Y¯i ∈ V (Ui ) to a tuple X V (S). One can think about Pi as an A-homomorphism from FR(S) into FR(Ui ) , then any solution ψ : FR(Ui ) → F pre-composed with Pi gives a solution φ : FR(S) → F ). Our elimination process can be viewed as a non-commutative analog of the classical elimination process in algebraic geometry. Hence, going “from the bottom to the top” every solution of the subsystem Sn = 1, . . . , Si = 1 can be extended to a solution of the next equation Si−1 = 1.

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Theorem 5 ([38], [39]). All solutions of the system of equations S = 1 in F (A) can be eﬀectively represented as homomorphisms from FR(S) into F (A) encoded into the following ﬁnite canonical Hom-diagram. Here all groups, except, maybe, the one in the root, are fully residually free, (given by a ﬁnite presentation) arrows pointing down correspond to epimorphisms (deﬁned eﬀectively in terms of generators) with non-trivial kernels, and loops correspond to automorphisms of the coordinate groups. σ1

qqq x qq q

FR(S) XXXX MMM XXXXXXX XXXXXX MMM XXXXXX & X, FR(Ωv2 ) FR(Ωvn ) ···

FR(Ωv1 ) σ2 MMM MMM uuu u u & zu FR(Ωv21 ) FR(Ωv2m ) ··· OOO oo OOO ooo OOO o o ' woo ··· ···

FR(Ωvk ) F (A) ∗ F (T ) F (A)

A family of homomorphisms encoded in a path from the root to a leaf of this tree is called a fundamental sequence or fundamental set of solutions (because each homomorphism in the family is a composition of a sequence of automorphisms and epimorphisms). Later Sela called such family a resolution. Therefore the solution set of the system S = 1 consists of a ﬁnite number of fundamental sets. And each fundamental set “factors through” one of the NTQ systems from Theorem 4. If S = 1 is irreducible, or, equivalently, G = FR(S) is fully residually free, then, obviously, one of the fundamental sets discriminates G. This gives the following result. Theorem 6 ([38], [39]). Finitely generated fully residually free groups are subgroups of coordinate groups of NTQ systems. There is an algorithm to construct an embedding. This corresponds to the extension theorems in the classical theory of elimination for polynomials. In [37] we have shown that an NTQ group can be embedded into a group obtained from a free group by a series of extensions of centralizers. Therefore Theorem 3 follows from Theorem 6. Since NTQ groups are fully residually free, fundamental sets corresponding to diﬀerent NTQ groups in Theorem 4 discriminate fully residually free groups

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which are coordinate groups of irreducible components of system S(X, A) = 1. This implies Theorem 7 ([38], [39]). There is an algorithm to ﬁnd irreducible components for a system of equations over a free group. Now we will formulate a technical result which is the keystone in the proof of Theorems 4 and 5. An elementary Abelian splitting of a group is the splitting as an amalgamated product or HNN-extension with Abelian edge group. Let G = A∗C B be an elementary Abelian splitting of G. For c ∈ C we deﬁne an automorphism φc : G → G such that φc (a) = a for a ∈ A and φc (b) = bc = c−1 bc for b ∈ B. If G = A∗C = A, t|ct = c , c ∈ C then for c ∈ C deﬁne φc : G → G such that φc (a) = a for a ∈ A and φc (t) = ct. We call φc a Dehn twist obtained from the corresponding elementary Abelian splitting of G. If G is an F -group, where F is a subgroup of one of the factors A or B, then Dehn twists that ﬁx elements of the free group F ≤ A are called canonical Dehn twists. If G = A ∗C B and B is a maximal Abelian subgroup of G, then every automorphism of B acting trivially on C can be extended to the automorphism of G acting trivially on A. The subgroup of Aut(G) generated by such automorphisms and canonical Dehn twists is called the group of canonical automorphisms of G. Let G and K be H-groups and A ≤ AutH (G) a group of H-automorphisms of G. Two H-homomorphisms φ and ψ from G into K are called A-equivalent (symbolically, φ ∼A ψ) if there exists σ ∈ A such that φ = σψ (i.e., g φ = g σψ for g ∈ G). Obviously, ∼A is an equivalence relation on HomH (G, K). Let G be a fully residually F group (F = F (A) ≤ G) generated by a ﬁnite set X (over F ) and A the group of canonical F automorphisms of G. Let F¯ = F (A∪Y ) a free group with basis A∪Y (here Y is an arbitrary set) and φ1 , φ2 ∈ HomF (G, F¯ ). We write φ1 < φ2 if there exists an automorphism σ ∈ A and an F -endomorphism π ∈ HomF (F¯ , F¯ ) such that φ2 = σ−1 φ1 π and |xφ1 | < |xφ2 |. x∈X

FR(S)

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An F -homomorphism φ : G → F¯ is called minimal if there is no φ1 such that φ1 < φ. In particular, if S(X, A) = 1 is a system of equations over F = F (A) and G = FR(S) then X ∪ A is a generating set for G over F . In this event, one can consider minimal solutions of S = 1 in F¯ . Deﬁnition 1. Denote by RA the intersection of the kernels of all minimal (with respect to A) F -homomorphisms from HomF (G, F¯ ). Then G/RA is called the maximal standard quotient of G and the canonical epimorphism η : G → G/RA is the canonical projection. Theorem 8 ([38]). The maximal standard quotient of a ﬁnitely generated fully residually free group is a proper quotient and can be eﬀectively constructed. This result (without the algorithm) is called the “shortening argument” in Sela’s approach.

5. Elimination process Given a system S(X) = 1 of equations in a free group F (A) one can eﬀectively construct a ﬁnite set of generalized equations Ω 1 , . . . , Ωk (systems of equations of a particular type) such that: • given a solution of S(X) = 1 in F (A) one can eﬀectively construct a reduced solution of one of Ωi in the free semigroup with basis A ∪ A−1 . • given a solution of some Ωi in the free semigroup with basis A ∪ A−1 one can eﬀectively construct a solution of S(X) = 1 in F (A). This is done as follows. First, we replace the system S(X) = 1 by a system of equations, such that each of them has length 3. This can be easily done by adding new variables. For one equation of length 3 we can construct a generalized equation as in the example below. For a system of equations we construct it similarly (see [38]). Example. Suppose we have the simple equation xyz = 1 in a free group. Suppose that we have a solution to this equation denoted by xφ , y φ , z φ where is φ is a given homomorphism into a free group F (A). Since xφ , y φ , z φ are reduced words in the generators A there must be complete cancellation. If we take a concatenation of the geodesic subpaths corresponding to xφ , y φ and z φ we obtain a path in the Cayley graph corresponding to this complete cancellation. This is called a cancellation −1 −1 φ tree. Then xφ = λ1 ◦ λ2 , y φ = λ−1 2 ◦ λ3 and z = λ3 ◦ λ1 , where u ◦ v denotes the product of reduced words u and v such that there is no cancellation between u and v. In the case when all the words λ1 , λ2 , λ3 are non-empty, the combinatorial generalized equation is shown at the bottom of Fig. 2. Given a generalized equation Ω one can apply elementary transformations (there are only ﬁnitely many of them) and get a new generalized equation Ω . If σ

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xyz = 1 O2

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Figure 2. From the cancellation tree for the equation xyz = 1 to the −1 −1 φ generalized equation (xφ = λ1 ◦ λ2 , y φ = λ−1 2 ◦ λ3 , z = λ3 ◦ λ1 ). is a solution of Ω, then elementary transformation transforms σ into σ . (Ω, σ) → (Ω , σ ). Elimination process is a branching process such that on each step one of the ﬁnite number of elementary transformations is applied according to some precise rules to a generalized equation on this step. Ω0 → Ω1 → · · · → Ωk . From the group theoretic view-point the elimination process tells something about the coordinate groups of the systems involved. This allows one to transform the pure combinatorial and algorithmic results obtained in the elimination process into statements about the coordinate groups. 5.1. Generalized equations Deﬁnition 2. A combinatorial generalized equation Ω (which is convenient to visualize as on the picture) 1

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1. A ﬁnite set of bases BS = BS(Ω). The set of bases M consists of 2n elements M = {μ1 , . . . , μ2n }. The set M comes equipped with two functions: a function ε : M → {1, −1} and an involution Δ : M → M (that is, Δ is a bijection such that Δ2 is an identity on M). Bases μ and μ are called dual bases. We denote bases by letters μ, λ, etc. 2. A set of boundaries BD = BD(Ω). BD is a ﬁnite initial segment of the set of positive integers BD = {1, 2, . . . , ρ + 1 + m}, where m is the cardinality of the basis A = {a1 , . . . , am } of the free group F = F (A). We use letters i, j, etc. for boundaries. (The example above has nine boundaries, ρ = 8, m = 0.) 3. Two functions α : BS → BD and β : BS → BD. We call α(μ) and β(μ) the initial and terminal boundaries of the base μ (or endpoints of μ). These functions satisfy the following conditions for every base μ ∈ BS: α(μ) < β(μ) if ε(μ) = 1 and α(μ) > β(μ) if ε(μ) = −1. (In the example α(λ) = 1, β(λ) = 4.) 4. The set of boundary connections (p, λ, q), where p is a boundary on λ (that is a number between α(λ) and β(λ)) and q is a boundary on λ. If (p, λ, q) is a boundary connection then (q, λ, p) is also a boundary connection. (The meaning of the boundary connections will be explained in ET5. To our example we can add some boundary connection, say (2, λ, 6). For the generalized equation to be consistent it is necessary that in the case ε(λ) = ε(λ), p1 > p2 implies q1 > q2 and in the case ε(λ) = −ε(λ), p1 > p2 implies q1 < q2 .) A boundary p is λ-tied if there is a boundary connection (p, λ, q) for some q. For a combinatorial generalized equation Ω, one can canonically associate a system of equations in variables h1 , . . . , hρ over F (A) (variables hi are sometimes called items). This system is called a generalized equation, and (slightly abusing the language) we denote it by the same symbol Ω. The generalized equation Ω consists of the following three types of equations. 1. Each pair of dual bases (λ, λ) provides an equation [hα(λ) hα(λ)+1 . . . hβ(λ)−1 ]ε(λ) = [hα(λ) hα(λ)+1 . . . hβ(λ)−1 ]ε(λ) . These equations are called basic equations. 2. Every boundary connection (p, λ, q) gives rise to a boundary equation [hα(λ) hα(λ)+1 · · · hp−1 ] = [hα(λ) hα(λ)+1 · · · hq−1 ], if ε(λ) = ε(λ) and [hα(λ) hα(λ)+1 · · · hp−1 ] = [hq hq+1 · · · hβ(λ)−1 ]−1 , if ε(λ) = −ε(λ). 3. Constant equations: hρ+1 = a1 , . . . , hρ+1+m = am . Remark 1. We assume that every generalized equation comes associated with a combinatorial one.

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Denote by FR(Ω) the coordinate group of the generalized equation. Deﬁnition 3. Let Ω(h) = {L1 (h) = R1 (h), . . . , Ls (h) = Rs (h)} be a generalized equation in variables h = (h1 , . . . , hρ ). A sequence of reduced nonempty words in an extended alphabet (A∪Z)±1 , U = (U1 (Z, A), . . . , Uρ (Z, A)) is a solution of Ω if: 1. all words Li (U ), Ri (U ) are reduced as written, 2. Li (U ) = Ri (U ), i ∈ [1, s]. If we specify a particular solution δ of a generalized equation Ω then we use a pair (Ω, δ). 5.2. Elementary transformations In this section we describe elementary transformations of generalized equations. Let Ω be a generalized equation. An elementary transformation (ET ) associates to a generalized equation Ω a family of generalized equations ET (Ω) = {Ω1 , . . . , Ωk } and surjective homomorphisms πi : FR(Ω) → FR(Ωi ) such that for any solution δ of Ω and corresponding epimorphism πδ : FR(Ω) → F there exists i ∈ {1, . . . , k} and a solution δi of Ωi such that the following diagram commutes. FR(Ω) πδ

- FR(Ωi )

πi

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? F ∗ F (Z) (ET1)

(ET2)

(ET3)

(Cutting a base (see Fig. 3)). Let λ be a base in Ω and p an internal boundary of λ with a boundary connection (p, λ, q). Then we cut the ¯ in q into the bases base λ in p into two new bases λ1 and λ2 and cut λ ¯ ¯ λ1 , λ2 . (Transferring a base (see Fig. 4)). If a base λ of Ω contains a base μ (that is, α(λ) ≤ α(μ) < β(μ) ≤ β(λ)) and all boundaries on μ are λtied by boundary connections then we transfer μ from its location on ¯ the base λ to the corresponding location on the base λ. ¯ (Removal of a pair of matched bases (see Fig. 5)). If the bases λ and λ ¯ ¯ ¯ are matched (that is, α(λ) = α(λ), β(λ) = β(λ)) then we remove λ, λ from Ω.

Remark 2. Observe, that for i = 1, 2, 3, we have k = 1, ET i(Ω) and Ω have the same set of variables H, and the identity map F [H] → F [H] induces an isomorphism π : FR(Ω) → FR(Ω ) . Moreover, δ is a solution of Ω if and only if δ is a solution of Ω . (ET4)

(Removal of a lone base (see Fig. 6)). Suppose, a base λ in Ω does not intersect any other base, that is, the items hα(λ) , . . . , hβ(λ)−1 are contained only on the base λ. Suppose also that all boundaries in λ are λ-tied, i.e.,

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for α(λ) ≤ i ≤ β(λ) − 1. It is not hard to see that π is an isomorphism. (Introduction of a boundary (see Fig. 7)). Suppose a boundary p in a base λ is not λ-tied. The transformation (ET5) λ-ties it. To this end, suppose δ is a solution of Ω. Denote λδ by Uλ , and let Uλ be the beginning of this word ending at p. Then we perform one of the following ¯ might be situtransformations according to where the end of Uλ on λ ated: ¯ is situated on the boundary q, we introduce (a) If the end of Uλ on λ the boundary connection p, λ, q. In this case the corresponding homomorphism πq : FR(Ω) → FR(Ωq ) is induced by the identity isomorphism on F [H]. Observe that θq is not necessary an isomorphism.

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Figure 4. Elementary transformation (ET2). ¯ is situated between q and q + 1, we in(b) If the end of Uλ on λ troduce a new boundary q between q and q + 1 (and rename all the boundaries); introduce a new boundary connection (p, λ, q ). Denote the resulting equation by Ωq . In this case the corresponding homomorphism πq : FR(Ω) → FR(Ωq ) is induced by the map πq (h) = h, if h = hq , and πq (hq ) = hq hq +1 . Observe that πq is an isomorphism. Obviously, the is only a ﬁnite number of possibilities such that for any solution δ one of them takes place. 5.3. Derived transformations and auxiliary transformations In this section we deﬁne complexity of a generalized equation and describe several useful “derived” transformations of generalized equations. Some of them can be realized as ﬁnite sequences of elementary transformations, others result in equivalent generalized equations but cannot be realized by ﬁnite sequences of elementary moves. A boundary is open if it is an internal boundary of some base, otherwise it is closed. A section is an interval σ = [i, . . . , i+ k]. It is said to be closed if boundaries i and i + k are closed and all the boundaries between them are open. Sometimes it will be convenient to subdivide all sections of Ω into active and non-active sections. Constant section will always be non-active. A variable hq is

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Figure 5. Elementary transformation (ET3). called free if it meets no base. Free variables are transported to the very end of the interval behind all items in Ω and become non-active. (D1) (Deleting a complete base). A base μ of Ω is called complete if there exists a closed section σ in Ω such that σ = [α(μ), β(μ)]. Suppose μ is a complete base of Ω and σ is a closed section such that σ = [α(μ), β(μ)]. In this case using (ET5), we transfer all bases from μ to μ ¯; using (ET4), we remove the lone base μ together with the section σ(μ). Complexity. Denote by n(σ) the number of bases in a closed section σ. The complexity of an equation Ω is the number τ = τ (Ω) = max{0, n(σ) − 2}, σ∈AΣΩ

where AΣΩ is the set of all active closed sections. (D2) (Linear elimination). Let γ(hi ) denote the number of bases met by hi . A base μ ∈ BS(Ω) is called eliminable if at least one of the following holds: (a) μ contains an item hi with γ(hi ) = 1, (b) at least one of the boundaries α(μ), β(μ) is diﬀerent from 1, ρ + 1, is not an endpoint of any other base, and is not connected by any boundary connection. We denote this boundary by . A linear elimination for Ω works as follows. Suppose the base μ is removable because it satisﬁes condition (b). We ﬁrst cut μ at the nearest to μ-tied boundary and denote it by τ . If there is no such

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Figure 6. Elementary transformation (ET4). a boundary we denote by τ the other boundary of μ. Then we remove the base obtained from μ between and τ together with its dual (maybe this part is the whole base μ), and remove the boundary . Denote the new equation by Ω . Suppose the base is removable because it satisﬁes condition (a). Suppose ﬁrst that γ(hi ) = 1 for the leftmost item hi on μ. Denote by the left boundary of hi . Let τ be the nearest to μ-tied boundary (or the other terminal boundary of μ if there are no μ-tied boundaries). We remove the base obtained from μ between and τ together with its dual (maybe this part is the whole base μ), and remove hi . We make a mirror reﬂection of this transformation if γ(hi ) = 1 for the rightmost item hi on μ. Suppose now that hi is not the leftmost or the rightmost item on μ. Let and τ be the nearest to hi μ-tied boundaries on the left and on the right of hi (each of them can be a terminal boundary of μ). We cut μ at the boundaries and τ , remove the base between and τ together with its dual and remove hi . Lemma 1. Linear elimination does not increase the complexity of Ω, and the number of items decreases. Therefore the linear elimination process stops after ﬁnite number of steps. Proof. The input of the closed sections not containing μ into the complexity does not change. The section than contained μ could be divided into two. The total number of bases can increase only if μ is eliminable according to case (a) and hi

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O

Figure 7. Elementary transformation (ET5). is not the leftmost or the rightmost item on μ. In this case the number of bases is increased by two, but the section is divided into two closed sections, and each section contains at least two bases. Therefore the complexity is the same. In all other cases the total number of bases does not increase, therefore the complexity cannot increase too. The number of items every time is decreased by one. We repeat linear elimination until no eliminable bases are left in the equation. The resulting generalized equation is called a kernel of Ω and we denote it by Ker(Ω). It is easy to see that Ker(Ω) does not depend on a particular linear elimination process. Indeed, if Ω has two diﬀerent eliminable bases μ1 , μ2 , and deletion of a part of μi results in an equation Ωi then by induction (on the number of eliminations) Ker(Ωi ) is uniquely deﬁned for i = 1, 2. Obviously, μ1 is still eliminable in Ω2 , as well as μ2 is eliminable in Ω1 . Now eliminating μ1 and μ2 from Ω2 and Ω1 we get one and the same equation Ω0 . By induction Ker(Ω1 ) = Ker(Ω0 ) = Ker(Ω2 ) hence the result. The following statement becomes obvious. Lemma 2. The generalized equation Ω (as a system of equations over F ) has a solution if and only if Ker(Ω) has a solution. So linear elimination replaces Ω by Ker(Ω). Let us consider what happens on the group level in the process of linear elimination. This is necessary only for the description of all solutions of the equation.

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We say that a variable hi belongs to the kernel (hi ∈ Ker(Ω)), if either hi belongs to at least one base in the kernel, or it is constant. Also, for an equation Ω by Ω we denote the equation which is obtained from Ω by deleting all free variables. Obviously, FR(Ω) = FR(Ω) ∗ F (Y¯ ) where Y¯ is the set of free variables in Ω. We start with the case when a part of just one base is eliminated. Let μ be an eliminable base in Ω = Ω(h1 , . . . , hρ ). Denote by Ω1 the equation resulting from Ω by eliminating μ. (a) Suppose hi ∈ μ and γ(hi ) = 1. Let μ = μ1 . . . μk , where μ1 , . . . , μk are the parts between μ-tied boundaries. Let hi ∈ μj . Replace the basic equation corresponding to μ by the equations corresponding to μ1 , . . . , μk . Then the variable hi occurs only once in Ω – precisely in the equation sμj = 1 corresponding to μj . Therefore, in the coordinate group FR(Ω) the relation sμj = 1 can be written as hi = w, where w does not contain hi . Using Tietze transformations we can rewrite the presentation of FR(Ω) as FR(Ω ) , where Ω is obtained from Ω by deleting sμj and the item hi . It follows immediately that FR(Ω1 )

FR(Ω ) ∗ hi

and FR(Ω)

FR(Ω )

FR(Ω1 ) ∗ F (B)

(1)

for some free or trivial group F (B). (b) Suppose now that μ satisﬁes case (b) in (D2) with respect to a boundary i. Let μ = μ1 . . . μk . Replace the equation sμ = 1 and the boundary equations corresponding to the boundary connections through μ by the equations sμi , i = 1, . . . , k. Then in the equation sμk = 1 the variable hi−1 either occurs only once or it occurs precisely twice and in this event the second occurrence ¯) is a part of the subword (hi−1 hi )±1 . In both cases it is easy to of hi−1 (in μ see that the tuple (h1 , . . . , hi−2 , sμk , hi−1 hi , hi+1 , . . . , hρ ) forms a basis of the ambient free group generated by (h1 , . . . , hρ ) and constants from A. Therefore, eliminating the relation sμk = 1, we can rewrite the presentation of FR(Ω) in generators Y¯ = (h1 , . . . , hi−2 , hi−1 , hi , hi+1 , . . . , hρ ). Observe also that any other basic or boundary equation sλ = 1 (λ = μ) of Ω either does not contain variables hi−1 , hi or it contains them as parts of the subword (hi−1 hi )±1 , that is, any such a word sλ can be expressed as a word wλ (Y¯ ) in terms of generators Y¯ . This shows that FR(Ω)

G(Y¯ )R(wλ (Y¯ )|λ =μ)

FR(Ω ) ,

where Ω is a generalized equation obtained from Ω1 by deleting the boundary i. Denote by Ω an equation obtained from Ω by adding a free variable z to

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O. Kharlampovich and A.G. Myasnikov the right end of Ω . It follows now that FR(Ω1 )

FR(Ω )

FR(Ω) ∗ z

and FR(Ω)

FR(Ω ) ∗ F (Z)

(2)

or some free group F (Z). Notice that all the groups and equations which occur above can be found eﬀectively. By induction on the number of steps in an elimination process we obtain the following lemma. Lemma 3. FR(Ω)

FR(Ker(Ω)) ∗ F (Z), where F (Z) is a free group on Z.

Proof. Let Ω = Ω0 → Ω1 → · · · → Ωl = Ker(Ω) be a linear elimination process for Ω. It is easy to see (by induction on l) that for every j ∈ [0, l − 1] Ker(Ωj ) = Ker(Ωj ). Moreover, if Ωj+1 is obtained from Ωj as in the case (b) above, then (in the notation above) Ker(Ωj )1 = Ker(Ωj ). Now the statement of the lemma follows from the remarks above and equalities (1) and (2). 5.4. Rewriting process for Ω In this section we describe a rewriting process for a generalized equation Ω. 5.4.1. Tietze cleaning and entire transformation. In the rewriting process of generalized equations there will be two main sub-processes: 1. Titze cleaning. This process consists of repetition of the following four transformations performed consecutively: (a) Linear elimination, (b) deleting all pairs of matched bases, (c) deleting all complete bases, (d) moving all free variables to the right and declare them non-active. 2. Entire transformation. This process is applied if γ(hi ) ≥ 2 for each hi in the active sections. We need a few further deﬁnitions. A base μ of the equation Ω is called a leading base if α(μ) = 1. A leading base is said to be maximal (or a carrier) if β(λ) ≤ β(μ), for any other leading base λ. Let μ be a carrier base of Ω. Any active base λ = μ with β(λ) ≤ β(μ) is called a transfer base (with respect to μ). Suppose now that Ω is a generalized equation with γ(hi ) ≥ 2 for each hi in the active part of Ω. An entire transformation is a sequence of elementary transformations which are performed as follows. We ﬁx a carrier base μ of Ω. We transfer all transfer bases from μ onto μ ¯. Now, there exists some i < β(μ) such that h1 , . . . , hi belong to only one base μ, while hi+1 belongs to at least two bases.

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Applying (ET1) we cut μ along the boundary i + 1. Finally, applying (ET4) we delete the section [1, i + 1]. Notice that neither process increases complexity. 5.4.2. Solution tree. Let Ω be a generalized equation. We construct a solution tree T (Ω) (with associated structures), as a rooted tree oriented from the root v0 , starting at v0 and proceeding by induction on the distance n from the root. Let v → v1 → · · · → vs → u be a path in T (Ω). If vi → vi+1 is an edge then there exists a ﬁnite sequence of elementary or derived transformations from Ωvi to Ωvi+1 and the epimorphism π(vi , vi+1 ) is the composition of the epimorphisms corresponding to these transformations. By π(v, u) we denote the composition of epimorphisms π(v, u) = π(v, v1 ) · · · π(vs , u). We also assume that active [non-active] sections in Ωvi+1 are naturally inherited from Ωvi , if not said otherwise. Suppose a path in T (Ω) is constructed by induction up to a level n, and suppose v is a vertex at distance n from the root v0 . We describe now how to extend the tree from v. We apply the Tietze cleaning at the vertex vn if it is possible. If it is impossible (γ(hi ) ≥ 2 for any hi in the active part of Ωv ), we apply the entire transformation. Both possibilities involve either creation of new boundaries and boundary connections or creation of new boundary connections without creation of new boundaries, and, therefore, addition of new relations to FR(Ωv ) . The boundary connections can be made in few diﬀerent ways, but there is a ﬁnite number of possibilities. According to this, diﬀerent resulting generalized equations are obtained, and we draw edges from v to all the vertices corresponding to these generalized equations. Termination condition: 1. Ωv does not contain active sections. In this case the vertex v is called a leaf or an end vertex. There are no outgoing edges from v. ¯ is oriented the opposite 2. Ωv is inconsistent. There is a base λ such that λ way and overlaps with λ, or the equation implies an inconsistent constant equation. 5.4.3. Quadratic case. Suppose Ωv satisﬁes the condition γi = 2 for each hi in the active part. Then FR(Ωv ) is isomorphic to the free product of a free group and a coordinate group of a standard quadratic equation (to be deﬁned below) over the coordinate group FR(Ω ) of the equation Ω corresponding to the non-active part. In this case entire transformation can go inﬁnitely along some path in T (Ω), and, since the number of bases if ﬁxed, there will be vertices v and w such that Ωv and Ωw are the same. Then the corresponding epimorphism π : FR(Ωv ) → FR(Ωw ) is an automorphism of FR(Ωv ) that decreases the total length of the interval. For a minimal solution of FR(Ωv ) the process will stop.

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Deﬁnition 4. A standard quadratic equation over the group G is an equation of the one of the following forms (below d, ci are nontrivial elements from G): n 1

[xi , yi ] = 1,

n > 0;

(3)

i=1 n 1

[xi , yi ]

i=1

m 1

zi−1 ci zi d = 1, n, m ≥ 0, m + n ≥ 1;

(4)

i=1 n 1

x2i = 1,

n > 0;

(5)

i=1 n 1 i=1

x2i

m 1

zi−1 ci zi d = 1, n, m ≥ 0, n + m ≥ 1.

(6)

i=1

Equations (3), (4) are called orientable, equations (5), (6) are called nonorientable. Number n is called a genus of the equation (notation gen(S).) The proof of the following fact can be found in [29]. Lemma 4. Let W be a strictly quadratic word over G (each variable occurs exactly twice). Then there is a G-automorphism f ∈ AutG (G[X]) such that W f is a standard quadratic word over G. 5.4.4. Entire transformation goes inﬁnitely. Let now γ(hi ) ≥ 2 for all hi in the active part, and for some hi the inequality is strict. Let (Ω, δ) be a generalized equation with a solution δ in the alphabet (A ∪ Z)±1 . A base participates in the entire transformation if it is a leading base or a transfer base. It is possible that the cleaning after the entire transformation decreases complexity. This occurs if some base is transferred onto its dual and removed by (ET3). Otherwise, we use the same name for a base of Ωi and the reincornation of this base in Ωi+1 . If we cannot apply Tietze cleaning after the entire transformation, then we successively apply entire transformation. It is possible that the entire transformation sequence for Ω goes inﬁnitely. Suppose the entire transformation goes inﬁnitely along some path in T (Ω), then after a ﬁnite number of steps, every base that participates, actually, participates inﬁnitely often. Deﬁne the excess ψ of (Ω, δ): ψ = Σλ (λδ ) − 2|I δ |, where λ runs through the set of bases participating in the entire transformation sequence and I = [h1 , . . . , hk ] is the segment between the initial boundary of the interval and the leftmost boundary k of the base that never participates (as carrier or transfer base). It is possible that the entire transformation goes inﬁnitely and the complexity does not decrease. If we apply the entire transformation to (Ω, δ) and the complexity does not decrease, then ψ does not change.

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We say that bases μ and its dual of the equation Ω form an overlapping pair if μ intersects with its dual μ ¯. If φ1 and φ2 are two solutions of a generalized equation Ω in F (A, Z), then we deﬁne φ1 2 for any hi ). Indeed, if σ is a quadratic section of Ω, we can cut all bases in Ω through the end-points of σ. Moreover, we will put the non-quadratic sections on the right part of the interval. Denote by Ω1 this new generalized equation. We will prove this theorem after proving key Lemmas 5-8. Lemma 5. If δ is a solution minimal with respect to the subgroup A generated by the canonical Dehn twists corresponding to the quadratic part of Ω, then one can construct a recursive function f = f (Ω) such that |I δ | ≤ f ψ. This lemma shows that for a minimal solution the length of the participating part of the interval is bounded in terms of the excess. And the excess does not change in the sequence of entire transformations when the complexity does not decrease. Proof. We apply the entire transformation to the pair (Ω1 , δ1 ), where δ1 is obtained from δ, and, therefore, minimal. We can ﬁnd a number k(Ω) such that after k transformations (Ω1 , δ1 ) → · · · → (Ωk , δk ) all bases situated on the quadratic part will either form matched pairs or will be transferred to the non-quadratic part. Indeed, while we transforming the quadratic part we notice that: 1) two equations Ωi and Ωj for i < j cannot be the same, because then δj would be shorter than δi , contradicting the minimality. 2) there is only a ﬁnite number of possibilities for the quadratic part since the number of items and complexity does not increase. The sequence of consecutive quadratic carrier bases is bounded. Therefore after a bounded number of steps, a quadratic coeﬃcient base is carrier, and we transfer a transfer base to the non-quadratic part. For a minimal solution, the length of a free variable corresponding to a matching pair is 1. And for each base λ

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transferred to the non-quadratic part, λδ is shorter than the interval corresponding to the non-quadratic part, and, therefore, shorter than ψ. This gives a hint how to compute a function f (Ω). We can now return to the generalized equation Ω and replace its solution by a minimal solution δ. The exponent of periodicity of a family of reduced words {w1 , . . . , wk } in a free group F is the maximal number t such that some wi contains a subword ut for some simple cyclically reduced word u. The exponent of periodicity of a solution δ is the exponent of periodicity of the family {hδ1 , . . . , hδρ }. We call a solution of a system of equations in the group F (A, Z) strongly minimal if it is minimal and cannot be obtained from a shorter solution by a substitution of reduced words from F (A, Z) instead of letters. Lemma 6 (Bulitko’s lemma). Let S be a system of equations over a free group. The exponent of periodicity of a strongly minimal solution can be eﬀectively bounded. Proof. Let P be a simple cyclically reduced word. A P -occurrence in a word w is an occurrence in w of a word P εt , ε = ±1, t ≥ 1. We call a P -occurrence v1 ·P εt ·v2 stable if v1 ends with P ε and v2 starts with P ε . Clearly, every stable P -occurrence lies in a maximal stable P -occurrence. Two distinct maximal stable P -occurrences do not intersect. A P -decomposition DP (w) of a word w is the unique representation of w as a product v0 · P ε1 r1 · v1 · · · · · P εm rm · vm where the occurrences of P εi ri are all maximal stable u-occurrences in w. If w has no stable P -occurrences then, by deﬁnition, its P -decomposition is trivial, that is, it has one factor which is w itself. By adding new variables we can transform the system S to the triangular form, namely, such that each equation has length 3. If we have equation xyz = 1 with solution xφ , y φ , z φ , then the cancellation table for this solution looks as the triangle in Fig. 2. Let xφ = v10 · P ε11 r11 · v11 · · · · · P ε1,m r1,m · v1,m , y φ = v20 · P ε21 r21 · u21 · · · · · P ε2,n r2,n · v2,n , z φ = v30 · P ε31 r31 · v31 · · · · · P ε3,k r3,k · v3,k be corresponding P -decompositions. From the cancellation table we will have a system of equations on the natural numbers rij , i = 1, 2, 3, j = 1, . . . , max{k, m, n}. All equations except, maybe, one will have form rij = rst for some pairs i, j and s, t and one equation may correspond to the middle of the triangle. If the middle of the triangle is inside a stable P -occurrence in z φ , then the equation would either have form r1j + r2s + 2 = r3t or r1j + r2s + 3 = r3t . Notice that since xφ , y φ , z φ are reduced words, the middle of the triangle cannot be inside a stable P -occurrence for more than one variable.

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If we replace a solution rij , i = 1,2,3, j = 1, . . . , max{k, m, n} of this system of equations by another positive solution, say qij , i = 1, 2, 3, j = 1, . . . , max{k, m, n} and replace in the solution xφ , y φ , z φ stable P -occurrences P rij by P qij we will have another solution of the equation xyz = 1. Now, instead of one equation xyz = 1 we take a system of equations S. We obtain a corresponding linear system for natural numbers rij ’s. Let R be the family of variables rij ’s that occur in the linear equations of length 3. The number of such equations is not larger than the number of triangles, that is the number of equations in the system S. Therefore R is a ﬁnite family. Consider a system of all linear equations on R. It depends on the particular solution of S, but there is a ﬁnite number of possible such systems. We now can replace values of variables from R by a minimal positive solution, say {qij }, of the same linear system (if rij does not appear in any linear equation we replace it by qij = 1) and replace in the solution of the system S stable P -occurrences P rij by P qij . We obtain another solution of the system S. The length of a minimal positive solution {qij } of the linear system is bounded as in the formulation of the lemma. The lemma is proved. Let G be a group, we say that H is a maximal fully residually free quotient of G if any other fully residually free quotient of G is a quotient of H. In particular, a fully residually free group G is the maximal fully residually free quotient of itself. Lemma 7. Suppose FR(Ω) is not a free product with an Abelian factor, and there are solutions of Ω with unboundedly large exponent of periodicity. One can eﬀectively ﬁnd a number M and a splitting with Abelian edge groups of a maximal fully residually free quotient of FR(Ω) as an amalgamated product with Abelian vertex group or as an HNN-extension (or both), such that the exponent of periodicity of a minimal solution of Ω with respect to the group generated by Dehn twists corresponding to this splitting and the quadratic part (if exists) is bounded by M . The proof of this lemma uses the notion of a periodic structure and can be found in ([41], Lemma 22) or in [38]. Consider an inﬁnite path in T (Ω) corresponding to an inﬁnite sequence in entire transformation r = v1 → v2 → · · · → vm . (7) Let δ be a solution of Ω. The following lemma gives the way to construct a function f1 depending on Ω such that for any number M , if the sequence of entire transformations for (Ω, δ) has f1 (M ) steps, then |I δ | > M ψ. Denote by μi the carrier base of the equation Ωvi . The path (7) will be called μ-reducing if μ1 = μ and one of the following holds: 1. μ2 does not overlap with its double and μ occurs in the sequence μ1 , . . . , μm−1 at least twice. 2. μ2 overlaps with its double and μ occurs in the sequence μ1 , . . . , μm−1 at least M + 2 times, where M is the exponent of periodicity of δ.

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O. Kharlampovich and A.G. Myasnikov The following lemma is just an easy exercise.

Lemma 8. In a μ-reducing path the length of I δ decreases at least by |μδ |/10. Proof. Case 1. μ = μ1 = μ2 , and not more than half of μ2 overlaps with its double. Then after two steps the leftmost boundary of the reincornation of μ will be to the right of the middle of μ2 . Therefore by the time when the reincornation of μ becomes a carrier, the part from the beginning of the interval to the middle of μ2 will be cut and removed. This part is already longer than half of μ. Case 2. μ = μ1 = μ2 , and μ2 (second reincornation of μ) does not overlap with its double. Then on the ﬁrst step we cut the part of the interval that is longer than half of μ. Case 3. μ2 overlaps with its double. Denote by μδ(i) the value of the reincornation of μδ on step i and by [1, σ]δ(i) the word corresponding to the beginning of the interval until boundary σ on step i. Then [1, α(¯ μ2 ))]δ(2) = P d for some cycliδ(2) cally reduced word P which is not a proper power and μδ(2) , μ2 are beginnings of [1, β(¯ μ2 )]δ(2) which is a beginning of P ∞ . We have μδ(2) = P r P1 , r ≤ M (8) Let μi1 = μi2 = μ for i1 < i2 and μi = μ for i1 < i < i2 . If δ(i +1)

1 |μi1 +1

and [1, ρi1 +1 + 1]

δ(i1 +1)

| ≥ 2|P |

(9) 3

begins with a cyclic permutation of P , then |[1, α(¯ μi1 +1 )]δ(i1 +1 | ≥ |P |.

The base μ occurs in the sequence μ1 , . . . , μm−1 at least r + 1 times, so either (9) fails for some i1 ≤ m − 1 or the part of the interval that was removed after m − 1 steps is longer than max{|r − 3||P |, |P |}. If (9) fails, then |[1, α(μi1 )]δ(i1 ) | ≥ (r − 2)|P |. So everything is reduced to the case when the part of the interval that was removed after m−1 steps is longer than max{|r − 3||P |, |P |}. Together with (8) this implies that in m − 1 steps the length 1 |μδ |. of the interval was reduced at least by 15 |μδ(2) | which is not less than 10 Proof of Theorem 9. Let L be the family of bases such that every base μ ∈ L occurs inﬁnitely often as a leading base. Suppose a number m is so big that for every base μ in L, a μ-reducing path occurs more than 20nf times during these m steps. Since Σ|μδm | ≥ ψ, where we sum over all the participating bases, at least for one base λ ∈ L, |λδm | ≥ ψ/2n. Moreover, |λδi | ≥ |λδm | ≥ ψ/2n for all i ≤ m. Since a λ-reducing path occurs more than 20nf times, the length of the interval would be decreased in m steps by more than it initially was. This gives a bound on the number of steps in the entire transformation sequence for (Ω, δ) for a minimal solution δ. Theorem 9 has been proved. Proof of Theorem 4. We replace in the tree T (Ω) every inﬁnite path corresponding to an inﬁnite sequence of entire transformation of generalized equations beginning

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at Ωvi by a loop corresponding to automorphisms of FR(Ωvi ) and ﬁnite sequence of transformations for a minimal solution of Ωvi . At the end of this sequence of transformations we obtain a generalized equation Ωvj such that either it has smaller complexity than Ωvi or FR(Ωvj ) is a proper quotient of FR(Ωvi ) . Any proper chain of residually free quotients is ﬁnite. Therefore we obtain a ﬁnite graph (the only cycles are loops corresponding to automorphisms) and its maximal subtree. The equation S(X) = 1 has a non-trivial solution if and only if we are able to construct Tsol (Ω) at least for one of the generalized equations corresponding to the system S(X) = 1. Let v0 → v1 → . . . vk be a path in Tsol (Ω) from the root to a leaf. Let vi−1 (i ≥ 1) be the ﬁrst vertex such that there is a loop corresponding to automorphisms of FR(Ωvi−1 ) attached to vi−1 . And let vj be the next such vertex or (if there is no such a second vertex) vj = vk . All the homomorphisms from FR(Ω) to F in the fundamental set corresponding to the path from v0 to vk factor through a free product of FR(Ωvi−1 ) and, maybe, some free group (that occurred as a result of Titze cleaning when going from Ωv0 to Ωvi−1 ). All the homomorphisms from FR(Ωvi−1 ) to F in the fundamental set corresponding to the path from vi−1 to vk are obtained by the composition of a canonical automorphism σ of FR(Ωvi−1 ) , canonical epimorphism π = πi . . . πj from FR(Ωvi−1 ) onto FR(Ωvj ) and a homomorphism from the fundamental set of homomorphisms from FR(Ωvj ) to F . The composition σπ is a solution of some system of equations, denoted by S1 (H1 , H2 , H3 , H π , A) = 1, over FR(Ωvj ) . (Notice that by H we denote a generating set of FR(Ωvi−1 ) modulo F (A)). Therefore H π and A are the sets of coeﬃcients of this system.) The system S1 (H1 , H2 , H3 , H π , A) = 1 consists of three types of subsystems: 1. Quadratic system in variables h ∈ H1 , where H1 is the collection of items in the quadratic part of Ωvi−1 . This system is obtained from Ω by replacing in each basic, and boundary equation each variable h in the non-quadratic part by the coeﬃcient hπ . 2. For each splitting of FR(Ωvi−1 ) as an amalgamated product with a free Abelian vertex group of rank k from the second part of Lemma 6, we reserve k variables x1 , . . . , xk ∈ H2 and write commutativity equations [xi , xj ] = 1 for i, j = 1, . . . , k and, in addition, equations [xi , uπ ] = 1 for each generator u of the edge group. 3. For each variable x ∈ H3 there is an equation xuπ x−1 = v π , where u is a generator of an edge group corresponding to a splitting of FR(Ωvi−1 ) as an HNNextension from the third part of Lemma 6 and x corresponds to the stable letter of this HNN-extension. Notice that σπ is a solution of the system S1 (H1 , H2 , H3 , H π , A) = 1 over the group FR(Ωvj ) . 4. For each x ∈ H3 and corresponding edge group we introduce a new variable y and equations [y, uπ ] = 1, where u is a generator of the edge group. Let H4 be the family of these new variables.

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Denote by S2 (H1 , H2 , H4 , H π , A) = 1 the system of equations 1,2 and 4. This system is NTQ over FR(Ωvj ) . Make a substitution x = xπ y. Using this substitution every solution σπ of the system S1 (H1 , H2 , H3 , H π , A) = 1 in the group FR(Ωvj ) can be obtained from a solution of the NTQ system S2 (H1 , H2 , H4 , H π , A) = 1. We made the induction step. Since Tsol (Ω) is ﬁnite, the proof of Theorem 4 can be completed by induction. It is clear that Theorem 9 is the main technical result required for the proof of Theorem 8 or, in other terminology, the base for our “shortening argument”. The proof of Theorem 9 is technically complicated because everything is done eﬀectively (the algorithms are given). For comparison we will give a non-constructive proof of Theorem 9 using the following lemma. Lemma 9. Let Ω0 , Ω1 , . . . be the generalized equations formed by the entire transformation sequence. Then one of the following holds. 1. the sequence ends, 2. for some i we obtain the quadratic case on the interval I, ¯ such that λ is a leading base, λδ begins 3. we obtain an overlapping pair λ, λ ¯ δ and there are solutions δ of Ωi with some nth power of the word [α(λ), α(λ)] with number n arbitrary large (with arbitrary large exponent of periodicity). Proof. We assume cases 1 and 2 do not hold. Then, our sequence is inﬁnite and we may assume that every base that is carried is carried inﬁnitely often, and that every base that carries does so inﬁnitely often. So every base that participates does so inﬁnitely often. We also assume the complexity does not change and that no base is moved oﬀ the interval. Let Ω be a generalized equation (with solution) and let B = Ω1 , . . . , Ωn , . . . be an inﬁnite branch. Let δ1 , δ2 , . . . be a set of solutions of Ω such that δi “factors” through Ωi . If we rescale the metric so that I δi has length 1, then each δi puts a length function on the items of Ω, in particular we assume that each base has length and midpoint between 0 and 1. This means that for each δi there is a point xi in [0, 1]m , where this point represents the lengths of the bases and the items as well as their midpoints in the normalized metric. We pass to a subsequence of δi (omit the double subscript) such that the xi converge to a point x in [0, 1]m . We call the limit a metric on (Ω, and denote it δ ∗ ). Normed excess denoted m(ψ) is a constant and we can apply the Bestvina, Feighn argument [7] (toral case) on the generalized equation Ω with lengths given by δ ∗ . The argument goes as follows. Entire transformation is moving bases to the right and shortening them, and m(ψ) is a constant. During the process the initial

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point of every base is only moved towards the ﬁnal point of I, and the length of a base is never increased, therefore, every base has a limiting position. Since m(ψ) is a constant, there is a base λ of length not going to zero that participates inﬁnitely often. If λ is eventually the only carrier, then we must have case 3 for the process to go unboundedly long. Suppose λ is carried inﬁnitely often. Whenever λ is the carrier, the midpoint of some base moves the distance between the midpoints of λ and its dual. Since every base has a limiting position, it follows that λ and its dual have the same limiting position. The argument shows that after some ﬁnite number of steps we get an overlapping initial section, i.e., carrier and dual have high length, but midpoints are close. It follows that for n suﬃciently large doing the process with (Ω, δn ) will give a similar picture. This implies case 3. Case 3 can only happen is there are solutions of an arbitrary large exponent of periodicity. If we consider only minimal solutions, then the exponent of periodicity can be eﬀectively bounded, and entire transformation always stops after a bounded number of steps. On the group level, case 2 corresponds to the existence a QH vertex group in the JSJ decomposition of FR(Ω) and case 3 corresponds to the existence of an Abelian vertex group in the Abelian JSJ decomposition of FR(Ω) .

6. Elementary free groups If an NTQ group does not contain non-cyclic Abelian subgroups we call it regular NTQ group. We have shown in [38] that regular NTQ groups are hyperbolic. (Later Sela called these groups hyperbolic ω-residually free towers [57].) Theorem 10. [44], [60] Regular NTQ groups are exactly the f.g. models of the elementary theory of a non-Abelian free group.

7. Stallings foldings and algorithmic problems A new technique to deal with F Z[t] became available when Myasnikov, Remeslennikov, and Serbin showed that elements of this group can be viewed as reduced inﬁnite words in the generators of F . It turned out that many algorithmic problems for ﬁnitely generated fully residually free groups can be solved by the same methods as in the standard free groups. Indeed, they introduces an analog of the Stallings’ folding for an arbitrary ﬁnitely generated subgroup of F Z[t] , which allows one to solve eﬀectively the membership problem in F Z[t] , as well as in an arbitrary ﬁnitely generated subgroup of it.

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Theorem 11 (Myasnikov-Remeslennikov-Serbin, [51]). Let G be a f.g. fully residually free group and G → G∗ the eﬀective Nielsen completion. For any f.g. subgroup H ≤ G one can eﬀectively construct a ﬁnite graph ΓH that in the group G∗ accepts precisely the normal forms of elements from H. Theorem 12 ([43]). The following algorithmic problems are decidable in a f.g. fully residually free group G: • the membership problem, • the intersection problem (the intersection of two f.g. subgroups in G is f.g. and one can ﬁnd a ﬁnite generated set eﬀectively), • conjugacy of f.g. subgroups, • malnormality of subgroups, • ﬁnding the centralizers of ﬁnite subsets. It was proved by Chadas and Zalesski [15] that ﬁnitely generated fully residually free groups are conjugacy separable. Notice that the decidability of conjugacy problem also follows from the results of Dahmani and Bumagin. Indeed, Dahmani showed that G is relatively hyperbolic with Abelian parabolics and Bumagin proved that the conjugacy problem is decidable in a relatively hyperbolic group provided it is solvable in parabolic subgroups. We prove that for ﬁnitely generated subgroups H, K of G there are only ﬁnitely many conjugacy classes of intersections H g ∩ K in G. Moreover, one can ﬁnd a ﬁnite set of representatives of these classes eﬀectively. This implies that one can eﬀectively decide whether two ﬁnitely generated subgroups of G are conjugate or not, and check if a given ﬁnitely generated subgroup is malnormal in G. Observe, that the malnormality problem is decidable in free groups, but is undecidable in torsion-free hyperbolic groups – Bridson and Wise constructed corresponding examples. We provide an algorithm to ﬁnd the centralizers of ﬁnite sets of elements in ﬁnitely generated fully residually free groups and compute their ranks. In particular, we prove that for a given ﬁnitely generated fully residually free group G the centralizer spectrum Spec(G) = {rank(C) | C = CG (g), g ∈ G}, where rank(C) is the rank of a free Abelian group C, is ﬁnite and one can ﬁnd it eﬀectively. Theorem 13 ([13]). The isomorphism problem is decidable in f.g. fully residually free groups. We also have an algorithm to solve equations in fully residually free groups and to construct the Abelian JSJ decomposition for them. Recently Dahmani and Groves [19] proved Theorem 14. The isomorphism problem is decidable in torsion-free relatively hyperbolic groups with Abelian parabolics. Dahmani [21] proved the decidability of the existential theory of a torsion free relatively hyperbolic group with virtually Abelian parabolic subgroups. This implies our result in [42] about the decidability of the existential theory of f.g. fully residually free groups.

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8. Residually free groups Any f.g. residually free group can be eﬀectively embedded into a direct product of a ﬁnite number of fully residually free groups [38]. Important steps towards the understanding of the structure of ﬁnitely presented residually free groups were recently made in [9, 10]. There exists ﬁnitely generated subgroups of F × F (this group is residually free but not fully residually free) with unsolvable conjugacy and word problem (Miller). In ﬁnitely presented residually free groups these problems are solvable [9]. Theorem 15 ([10]). Let G < Γ0 × · · · × Γn be the subdirect product of limit groups. Then G is ﬁnitely presented iﬀ it satisﬁes the virtual surjection to pairs (VSP) property: ∀ 0 ≤ i < j ≤ n |Γi × Γj : Pij (G)| < ∞.

References [1] K.I. Appel. One-variable equations in free groups. Proc. Amer. Math. Soc., 19:912– 918, 1968. [2] G. Baumslag, On generalized free products, Math. Z., 78:423–438, 1962. [3] B. Baumslag, Residually free groups, Proc. London. Math. Soc. (3), 17:402–418, 1967. [4] J.L. Bell, A.B. Slomson, Models and ultraproducts: an introduction, North-Holland, Amsterdam, 1969. [5] G. Baumslag, A. Myasnikov, V. Remeslennikov. Algebraic geometry over groups I. Algebraic sets and ideal theory. Journal of Algebra, 1999, v. 219, 16–79. [6] G. Baumslag, A. Myasnikov and V. Remeslennikov, Malnormality is decidable in free groups. Internat. J. Algebra Comput. 9 no. 6 (1999), 687–692. [7] M. Bestvina, M. Feighn, Stable actions of groups on real trees, Invent. Math., 1995, v. 121, 2, pp. 287–321. [8] Bestvina, M.; Feighn, M. A combination theorem for negatively curved groups. J. Diﬀerential Geom. 35 (1992), no. 1, 85–101. Addendum and correction to: “A combination theorem for negatively curved groups” J. Diﬀerential Geom. 35 (1992), no. 1, 85–101, J. Diﬀerential Geom. 43 (1996), no. 4, 783–788. [9] M. Bridson, J. Howie, C. Miller, H. Short, Subgroups of direct products of limit groups, arXiv:0704.3935v2, 6Nov 2007, Annals of Math., in press. [10] M. Bridson, J. Howie, C. Miller, H. Short, Finitely presented residually free groups, arXiv:0809.3704v1, 22 Sep. 2008. [11] Bryant R., The verbal topology of a group, Journal of Algebra, 48:340–346, 1977. [12] I. Bumagin, The conjugacy problem for relatively hyperbolic groups. Algebr. Geom. Topol. 4 (2004), 1013–1040. [13] I. Bumagin, O. Kharlampovich, A. Myasnikov. Isomorphism problem for ﬁnitely generated fully residually free groups., J. Pure and Applied Algebra, Volume 208, Issue 3, March 2007, Pages 961–977. [14] C.C. Chang, H.J. Keisler, Model Theory. North-Holland, London, N.Y., 1973.

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[15] S.C. Chagas, P. Zalesskii, Limit Groups are Conjugacy Separable. IJAC 17(4): 851– 857 (2007) [16] C. Champetier, V. Guirardel, Limit groups as limits of free groups: compactifying the set of free groups, Israel Journal of Mathematics 146 (2005), 1–75. [17] D.E. Cohen, Combinatorial group theory: a topological approach. Cambridge Univ. Press, 1989. [18] L.P. Comerford Jr. and C.C. Edmunds. Solutions of equations in free groups. Walter de Gruyter, Berlin, New York, 1989. [19] F. Dahmani, D. Groves. The isomorphism problem for toral relatively hyperbolic ´ groups. Publ. Math. Inst. Hautes Etudes Sci. No. 107 (2008), 211–290. [20] F. Dahmani, Existential questions in (relatively) hyperbolic groups. to appear in Israel J. Math. [21] F. Dahmani, Combination of convergence groups. Geom. Topol. 7 (2003), 933–963. [22] Yu.L. Ershov, E.A. Palutin, Mathematical Logic. Walter de Gruyter, Berlin, New York, 1989. [23] B. Fine, A.M. Gaglione, A. Myasnikov, G. Rosenberger, and D. Spellman. A classiﬁcation of fully residually free groups of rank three or less. Journal of Algebra 200 (1998), no. 2, 571–605. MR 99b:20053 [24] V. Guirardel, Limit groups and groups acting freely on Rn -trees, Geom. Topol. 8 (2004), 1427–1470. [25] V. Guba, Equivalence of inﬁnite systems of equations in free groups and semigroups to ﬁnite subsystems. Mat. Zametki, 40:321–324, 1986. [26] D. Gildenhuys, O. Kharlampovich, and A. Myasnikov, CSA groups and separated free constructions. Bull. Austr. Math. Soc., 1995, 52, 1, pp. 63–84. [27] R.I. Grigorchuk and P.F. Kurchanov. On quadratic equations in free groups. Contemp. Math., 131(1):159–171, 1992. [28] D. Groves, H. Wilton. Enumerating limit groups, arXiv:0704.0989v2. [29] R.C. Lyndon and P.E. Schupp. Combinatorial group theory. Springer, 1977. [30] R.C. Lyndon. Groups with parametric exponents. Trans. Amer. Math. Soc., 96:518– 533, 1960. [31] R.C. Lyndon. Equations in free groups. Trans. Amer. Math. Soc. 96 (1960), 445–457. [32] R.C. Lyndon. Equations in groups. Bol. Soc. Bras. Mat., 11:79–102, 1980. [33] R.C. Lyndon, Groups with parametric exponents, Trans. Amer. Math. Soc., 96, 518– 533, (1960). [34] A.I. Malcev, On equation zxyx−1 y −1 z −1 = aba−1 b−1 in a free group, Algebra and Logic, 1 (1962), 45–50. [35] A. Myasnikov, V. Remeslennikov, Degree groups, Foundations of the theory and tensor completions, Sibirsk. Mat. Zh., 35, (1994), 5, 1106–1118. [36] A. Myasnikov, V. Remeslennikov, Algebraic geometry over groups II: Logical foundations, J. Algebra, 234 (2000), pp. 225–276. [37] O. Kharlampovich and A. Myasnikov. Irreducible aﬃne varieties over a free group. 1: irreducibility of quadratic equations and Nullstellensatz. J. of Algebra, 200:472–516, 1998. MR 2000b:20032a

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[38] O. Kharlampovich and A. Myasnikov, Irreducible aﬃne varieties over a free group. II: Systems in triangular quasi-quadratic form and description of residually free groups. J. of Algebra, v. 200, no. 2, 517–570, 1998. MR 2000b:20032b [39] O. Kharlampovich and A. Myasnikov. Description of Fully Residually Free Groups and Irreducible Aﬃne Varieties Over a Free Group. Banﬀ Summer School 1996, Centre de Recherches Math´ematiques, CRM Proceedings and Lecture Notes, v. 17, 1999, p. 71–80. MR 99j:20032 [40] O. Kharlampovich, A. Myasnikov, Hyperbolic groups and free constructions, Trans. Amer. Math. Soc. 350 (1998), no. 2, 571–613. [41] O. Kharlampovich, A. Myasnikov, Implicit function theorems over free groups. J. Algebra, 290 (2005) 1–203. [42] O. Kharlampovich, A. Myasnikov, Eﬀective JSJ decompositions, Group Theory: Algorithms, Languages, Logic, Contemp. Math., AMS, 2004, 87–212 (Math GR/0407089). [43] O. Kharlampovich, A. Myasnikov, V. Remeslennikov, D. Serbin. Subgroups of fully residually free groups: algorithmic problems,Group theory, Statistics and Cryptography, Contemp. Math., Amer. Math. Soc., 360, 2004, 61–103. [44] Kharlampovich O., Myasnikov A., Elementary theory of free non-abelian groups, J. Algebra, 302, Issue 2, 451–552, 2006. [45] O. Kharlampovich, A. Myasnikov, D. Serbin, Groups with free regular length function on Zn . [46] O. Kharlampovich, A. Myasnikov, D. Serbin Zn -free groups. [47] E. Lioutikova, Lyndon’s group is conjugately residually free. Internat. J. Algebra Comput. 13 (2003), no. 3, 255–275. [48] G.S. Makanin. Equations in a free group (Russian). Izv. Akad. Nauk SSSR, Ser. Mat., 46:1199–1273, 1982. transl. in Math. USSR Izv., V. 21, 1983; MR 84m:20040. [49] G.S. Makanin. Decidability of the universal and positive theories of a free group (Russian). Izv. Akad. Nauk SSSR, Ser. Mat., 48(1):735–749, 1985. transl. in Math. USSR Izv., V. 25, 1985; MR 86c:03009. [50] Ju.I. Merzljakov. Positive formulae on free groups. Algebra i Logika, 5(4):25–42, 1966. [51] Myasnikov A., Remeslennikov V., Serbin D., Fully residually free groups and graphs labeled by inﬁnite words. to appear in IJAC. [52] P. Pfander, Finitely generated subgroups of the free Z[t]-group on two generators, Model theory of groups and automorphism groups (Blaubeuren, 1995), 166–187, London Math. Soc. Lecture Note Ser., 244, Cambridge Univ. Press, Cambridge, 1997. [53] E. Rips and Z. Sela. Cyclic splittings of ﬁnitely presented groups and the canonical JSJ decomposition. Annals of Math., 146, 53–109, 1997. [54] A. Razborov. On systems of equations in a free group. Math. USSR, Izvestiya, 25(1):115–162, 1985. [55] A. Razborov. On systems of equations in a free group. PhD thesis, Steklov Math. Institute, Moscow, 1987. [56] V. Remeslennikov, ∃-free groups, Siberian Math J., 30 (6):998–1001, 1989.

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[57] Z. Sela. Diophantine geometry over groups I: Makanin-Razborov diagrams. Publications Math´ematiques de l’IHES 93(2001), 31–105. [58] J.R. Stallings. Finiteness of matrix representation. Ann. Math., 124:337–346, 1986. [59] L. Ribes, P. Zalesskii, Conjugacy separability of amalgamated free products of groups. J. Algebra, 1996, v. 179, 3, pp. 751–774 [60] Z. Sela. Diophantine geometry over groups VI: The elementary theory of a free group. GAFA, 16(2006), 707–730. [61] N. Touikan, On the coordinate groups of irreducible systems of equations in two variables over free groups, arXiv:0810.1509v3, 11 Nov. 2008. Olga Kharlampovich and Alexei G. Myasnikov McGill University Department of Mathematics and Statistics Burnside Hall, room 915 805 Sherbrooke West, Montreal Quebec, Canada, H3A 2K6 e-mail: [email protected] [email protected]

Combinatorial and Geometric Group Theory Trends in Mathematics, 243–250 c 2010 Springer Basel AG

The FN -action on the Product of the Two Limit Trees for an Iwip Automorphism Martin Lustig Abstract. An elementary proof is given for the fact that, for every non-surface iwip automorphisms ϕ of a free group FN , the FN -action, on the cartesian product T+ (ϕ) × T+ (ϕ−1 ) of the (non-simplicial) forward limit R-trees for ϕ and ϕ−1 , is properly discontinuous. Alternative proofs, derived from deeper results, have been given by Bestvina-Feighn-Handel [3] and later by LevittLustig [10]; compare also Guirardel [9]. Mathematics Subject Classiﬁcation (2000). Primary 20F36, Secondary 20E36, 57M05. Keywords. R-trees, discrete action on product, iwip automorphisms of free groups.

1. Introduction Let ϕ ∈ Aut(FN ) be an irreducible outer automorphism, with irreducible positive powers, of a ﬁnitely generated non-abelian free group FN (see §2). We also assume ϕ to be non-geometric, i.e., ϕ is not induced by a homeomorphism of a surface with boundary. Then there exists an R-tree T+ (ϕ) with isometric FN -action that has the following properties (compare §2): – The FN -action on T is free, but not simplicial: the FN -orbit of every point is dense in T+(ϕ). – The tree T+(ϕ) is projectively invariant under ϕ, with stretching factor λ > 1, and T+ (ϕ) is obtained from iterating some train track representative f : τ → τ of ϕ. – The tree T+ (ϕ) is uniquely determined by ϕ up to FN -equivariant homothety. Analogous statements hold for ϕ−1 as well. Theorem 1.1. The product action of FN on T+ (ϕ) × T+ (ϕ−1 ) is free and properly discontinuous.

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The ﬁrst proofs of this result appeared in [3] and [12]. In [10] and [9] more general results have been proved. The proof given here, a slightly elaborate version of [12], is short and elementary (in the sense that it only uses standard train track techniques). More references and background are given below in the last section. To prove this theorem we show that large powers of ϕ contract any conjugacy class [w] ⊂ FN , provided that w has suﬃciently small T+ (ϕ)-translation length ||w|| = min{d(x, wx) | x ∈ T+ (ϕ)}. More precisely: Proposition 1.2. For any basis A of FN there exists a number r(ϕ) ≥ 1 with the property that for any integer r ≥ r(ϕ) there is a bound K(r) > 0 such that, for any w ∈ FN with T+ (ϕ)-translation length ||w|| ≤ K(r), the lengths (with respect r (w) satisfy the strict inequality to A) of the cyclically reduced words w and ϕ r (w)| < |w| |ϕ A. A This proposition implies directly the above Theorem 1.1, as, for r ≥ r(ϕ), r(ϕ−1 ), if w (and hence ϕr (w)) had small translation length on both, T+ (ϕ) and T+ (ϕ−1 ), we would get the contradiction r (w)| < |w| |w| A = | ϕ−r (ϕr (w)) |A < |ϕ A. A

2. The set-up In this section we recall some known facts about train track representatives and the limit tree T+ (ϕ). This also serves to ﬁx our notation. The reader who prefers a more expanded version of the presentation given in this section is referred to §§2–5 of [13]. An automorphism ϕ ∈ Aut(FN ) is irreducible with irreducible powers (iwip) if no ϕt , for any t ≥ 1, ﬁxes a proper free factor of FN up to conjugation. Every such ϕ is represented by a train track map f : τ → τ with associated geometric transition matrix M (f ) = (me,e )e,e ∈Edges(τ ) , where the coeﬃcient me,e denotes the number of times that the edge path f (e ) crosses over the edge e or its inverse (both counted positively). The integer matrix M (f ) is non-negative and irreducible (in the standard meaning for non-negative matrices), and thus has Perron-Frobenius eigenvalue (= the spectral radius of M (f )) λ > 1, see [1]. A path γ in τ is called reduced if there is no backtracking along γ (i.e., if γ, interpreted as map from part of R to τ , is locally injective). For any non-reduced γ we denote by [γ] the reduced path resulting from γ by cancelling all backtracking subpaths. Notice that one always has [f (γ)] = [f ([γ])] .

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A reduced path γ is called legal if f t (γ) is reduced, for all t ≥ 1. In particular, every edge of τ is legal; in fact, this is precisely the deﬁning property of a train track map. If γ (possibly non-reduced) is not legal, then a point of γ (always a vertex of τ ) which separates two maximal legal subpaths of γ is called an illegal turn in γ. We denote by ILT(γ) the number of illegal turns in γ. If γ is a closed path in τ , then by “reduced” or “legal” we always mean cyclically reduced or cyclically legal. Similarly, [ γ ] denotes the cyclically reduced path, and ILT( γ ) denotes the number of illegal turns on the cyclic path γ. Fact 2.1. The R-tree T+ (ϕ), obtained from iterating some train track representative f : τ → τ of ϕ, is projectively invariant: ||ϕ(w)|| = λ ||w||

for all w ∈ FN ,

with λ > 1 as given above (see [2], [8], [11]). For the universal covering τ of τ there is an FN -equivariant map i : τ → T+ (ϕ). One can deﬁne a PF-length for every edge of τ , given through an eigenvector v of the Perron-Frobenius eigenvalue λ > 1 of M (f ). By an edge path we mean a path which starts and ends in a vertex. Two edge paths are equal if the sequences of edges traversed, with orientation, are the same. Most paths in this papers are edge paths, even if this is not explicitly stated. Fact 2.2. For every legal path γ in τ , one has: PF-length(f (γ)) = λ PF-length(γ). Fact 2.3. For every legal path γ in τ , and any lift γ of γ to τ , one has: length(i(γ)) = PF-length(γ) . If γ is a (not necessarily legal) edge path in τ we denote by the simplicial length of γ the number of edges which are crossed by γ. As M (f ) is irreducible, it follows that all entries of the eigenvector v are strictly positive. Hence the two lengths are related as follows: Fact 2.4. There exists a constant C > 0 such that C −1 · PF-length(γ) ≤ simplicial-length(γ) ≤ C · PF-length(γ) . Fact 2.5. For every w ∈ FN , and every reduced loop γ in τ which represents the conjugacy class of w, one has: PF-length([f s (γ)] ||w|| = lim . s→∞ λs From the ﬁniteness of τ and the fact that f represents an automorphism one deduces: Fact 2.6. There is an upper bound B ≥ 0 for the PF-length of a backtracking path in the image f (γ) of any reduced path γ in τ (compare [8]).

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Fact 2.7. An irreducible ϕ ∈ Aut(FN ) with irreducible powers is geometric (= induced by a surface homeomorphism) if and only if there is a non-trivial ϕperiodic conjugacy class in FN (see [1], §4).

3. The proof of Proposition 1.2 In order to prove Proposition 1.2 we now derive two lemmas: Lemma 3.1. There are constants K > 0 and ε > 0 such that every closed reduced path γ in τ with legal subpath γ0 of simplicial length bigger than K represents the conjugacy class of an element w ∈ FN with # w #> ε. Proof. Let γ0 be a (non-closed) legal path in τ with non-trivial subpath γ1 , such that both components γ(i) (for i = 1, 2) of γ0 γ1 satisfy (λ − 1) PF-length(γ(i)) ≥ B for B as in Fact 2.6. Calling such a pair (γ0 , γ1 ) B-long, we observe, by Fact 2.2, that the pair of legal paths (f (γ0 )∗ , f (γ1 )) is again B-long, where f (γ0 )∗ denotes the path obtained from f (γ0 ) after cancelling boundary subpaths of PF-length B. Hence, by Facts 2.2, 2.5 and 2.6, if the conjugacy class of any element w ∈ FN is represented by a reduced loop γ in τ with a B-long pair of subpaths (γ0 , γ1 ), then one has ||w|| ≥ PF-length(γ1 ) > 0. Thus the statement of Lemma 3.1 follows from Fact 2.4. For K as in Lemma 3.1 we denote by L(t) the set of closed or non-closed reduced edge paths γ in τ such that [f t (γ)] does not contain any legal subpath of simplicial length bigger than K, for all 0 ≤ t ≤ t, with the possible exception, for t ≥ 1, of the initial and terminal maximal legal subpath, in case γ is non-closed. Notice that by deﬁnition one has · · · ⊂ L(t + 1) ⊂ L(t) ⊂ · · · ⊂ L(0) . If γ is a subpath of a reduced path γ, we consider the subpath f (γ ) of f (γ), and the corresponding reduced paths [f (γ )] and [f (γ)]. It is possible that [f (γ )] is not a subpath of [f (γ)] but bifurcates from [f (γ)] at initial and terminal subpaths. We denote by [f (γ )]γ the subpath of [f (γ )] obtained by erasing these initial and terminal subpaths (admitting the degenerate cases where the leftover path [f (γ )]γ consists of a single point only or is empty), and we note that, if γ is a concatenation γ = γ1 γ2 . . . γr of reduced subpaths, it follows that [f (γ)] is the concatenation [f (γ)] = [f (γ1 )]γ [f (γ2 )]γ . . . [f (γr )]γ . Lemma 3.2. There exists a constant C ≥ 1 and an exponent q ≥ 1 with the following properties: (a) Every subpath γ of an edge path γ ∈ L(q), with ILT(γ ) = C, satisﬁes ILT([f q (γ )]γ ) < ILT(γ ).

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(b) For any t ≥ 1, if γ ∈ L(tq) and ILT(γ) ≥ 2t C, then

t 2C − 1 ILT(γ) . ILT([f tq (γ)]) ≤ 2C Proof. (a) We ﬁrst notice that for any constant K > 0 there exist only ﬁnitely many distinct legal edge paths γk in τ with simplicial-length(γk ) ≤ K (recall that two edge paths are equal iﬀ the sequences of edges traversed, with orientation, are the same). It follows that there is a bound C ≥ 1 such that, if γ ∈ L(0) and γ is a subpath of γ with ILT(γ ) = C, written as concatenation of maximal legal subpaths γ = γ1 γ2 . . . γC+1 , then there exist indices i < j in {1, . . . , C} with γi = γj and γi+1 = γj+1 . Furthermore, it follows that there are only ﬁnitely many closed edge paths γ ∈ L(0) with ILT( γ ) ≤ C. Now, the fact that the f -image of every legal path is again legal implies that the number of illegal turns in any path in τ will not be increased when applying f . Let us assume that, for γ and γ as in the previous paragraph, and for some q ≥ 1, one has γ ∈ L(q) and ILT([f q (γ )]γ ) = ILT(γ ). It follows that for none of the maximal legal subpath γi of γ the subpath f q (γi ) of f q (γ ) degenerates to a single point or is cancelled entirely when passing from f q (γ ) to [f q (γ )]γ . In other words, every maximal legal subpath γi of γ contains a point with f q -image that does not lie on one of the backtracking subpaths of f q (γ) at the two endpoints of f q (γi ). Hence, for any two adjacent maximal legal subpaths γi , γi+1 of γ , the backtracking subpath in f q (γ) at the illegal turn which connects f q (γi ) to f q (γi+1 ) depends only on γi and γi+1 , and not on the rest of the path γ. In particular, as γ = γ1 γ2 . . . γC+1 with γi = γj and γi+1 = γj+1 for 1 ≤ i < j ≤ C, we obtain a closed edge path γ , deﬁned by cyclic concatenation of the path γi+1 γi+2 . . . γj , which satisﬁes γ ∈ L(q) and ILT(f t ( γ )) = ILT( γ ) = j − i ≤ C for 1 ≤ t ≤ q. As the number of closed edge paths in L(0) with C or less illegal turns is ﬁnite, it follows that, if q is suﬃciently big, the conjugacy classes given by the γ ) cannot be pairwise distinct for all 1 ≤ t ≤ q. Hence γ determines a non-trivial f t ( conjugacy class in FN which is ϕ-periodic. But this contradicts our assumption that f represents a non-geometric irreducible automorphism with irreducible powers, see Fact 2.7. Hence there must be an upper bound q0 ≥ 1 to all exponents q ≥ 1 with ILT([f q (γ )]γ ) = ILT(γ ) for any subpath γ of a path γ ∈ L(q) with ILT(γ ) = C. Thus any q > q0 satisﬁes the claim (a). (b) Let q and C be as in part (a) and let γ be a path in L(q) with ILT(γ) ≥ C. We can subdivide γ at illegal turns into edge paths γi with 2C ≥ ILT(γi ) ≥ C, and apply (a) to deduce that ILT([f q (γi )]γ ) ≤ ILT(γi )−1 for each γi . As γ decomposes 1 into at least 2C ILT(γ) of such γi , this gives ILT([f q (γ)]) ≤ ILT(γ) −

2C − 1 ILT(γ) = ILT(γ) . 2C 2C

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Thus, if γ is a path in L(tq) for some t ≥ 1 and ILT(γ) ≥ 2t C, then either one has ILT([f t q (γ)]) < C for some 1 ≤ t ≤ t, in which case it follows t

t

1 2C − 1 tq t q ILT(γ) ≤ ILT(γ) . ILT([f (γ)]) ≤ ILT([f (γ)]) < C ≤ 2 2C

Otherwise, we can apply the above consideration t times to γt = [f t q (γ)], for t = t, t − 1, . . . , 1, thus computing 2C − 1 ILT([f (t−1)q (γ)]) ILT([f tq (γ)]) ≤ 2C 2 t

2C − 1 2C − 1 ≤ ILT([f (t−2)q (γ)]) ≤ · · · ≤ ILT(γ) . 2C 2C This proves our assertion. Proof of the proposition. For any basis A of FN there is a constant d > 0 such that the A-length of any cyclically reduced word w ∈ FN and the simplicial length of the corresponding closed reduced path γ in τ are related by inequalities d −1 |w| A ≤ simplicial-length( γ ) ≤ d |w| A. Furthermore, for K > 0 as in Lemma 3.1 and γ ∈ L(0) one has the inequalities K −1 simplicial-length( γ ) ≤ ILT( γ ) ≤ simplicial-length( γ) . Hence there is a constant D > 0 such that A ≤ ILT( γ ) ≤ D |w| A. D−1 |w| 2C Now, choose t large enough such that ( 2C−1 )t > D 2 , for C as in Lemma 3.2. For any r ≥ tq we ﬁnd, by Lemma 3.1 and Fact 2.1, a constant ε(r) > 0 such that the condition ||w|| ≤ ε(r) implies γ ∈ L(r), and furthermore ILT( γ ) ≥ 2t C, as the number of non-trivial closed edge paths γ in L(0) with ILT( γ ) < 2t C is ﬁnite. But then we obtain, using Lemma 3.2 (b), the inequalities

r (w)| ≤ D · ILT([f r ( |ϕ γ )]) ≤ D · ILT([f tq ( γ )]) A t

2C − 1 ILT( γ ) < D−1 · ILT( γ ) ≤ |w| A. ≤D· 2C

4. A little history and some references The statement of Theorem 1.1 was originally conjectured by Peter Shalen (in the early ’90s) and communicated to me by Gilbert Levitt, who, in joint work with Culler and Shalen [6], investigated consequences of the analogous result for pseudo-Anosov homeomorphisms of closed surfaces. Around the time that the ﬁrst written version [12] of the proof presented here was circulated, Bestvina, Feighn and Handel had obtained an independent proof as consequence of their deep work on the Tits alternative in Out(FN ), see Theorem 5.3 of [3]. The advantage of the proof presented in the preceding sections is that it is short and direct in that

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it uses only standard facts about train track maps, essentially known since their introduction in [1]. Our result is also slightly more general than Theorem 5.3 of [3], as it does not just apply to irreducible automorphisms with irreducible powers (compare [4]). The precise conditions for this generalization are given as follows: Remark 4.1. The statement of Proposition 1.2 and hence that of Theorem 1.1 are also true for a more general class of automorphisms of FN . The proof given in §3 uses only the weaker assumption that there exists a train track representative f : τ → τ of ϕ with transition matrix M (f ) which admits a strictly positive row eigenvector v with eigenvalue strictly bigger than 1, and that there is no ϕ-periodic conjugacy class in FN {1}. Examples of such automorphisms, which however are not iwip, are given for example by attaching an extra edge e to a ﬁxed point P of a train track map f that represents a non-geometric iwip automorphism, if one extends f by deﬁning f (e) = eγ, for any non-trivial legal loop γ with endpoints at the ﬁxed point P . Shalen stated his original conjecture in the context of the question whether the groups FN Z are coherent (meaning: “ﬁnitely generated subgroups are ﬁnitely presented”), in analogy to 3-manifold groups. This question has been answered in the positive by Feighn-Handel [7], but without using the above Theorem 1.1. However, apart from its natural appeal within the theory of group actions on Rtrees, the proof presented here seems also to be of some interest on is own: for example, Peter Brinkmann told me that it inspired him for a crucial part of his thesis (compare [5]), and the main idea employed in §3 is also used again in [14]. For arbitrary automorphisms of FN a strict generalization of Theorem 1.1 is clearly wrong. However, using methods related to the above proof, but in a much more technically loaded context, the following result has been proved in [10]: Theorem 4.2 (Levitt-Lustig). For every ϕ ∈ Aut(FN ) there exist projectively invariant R-trees T+ and T− , with stretching factors λ+ ≥ 1 and λ− ≤ 1 respectively, as well as a constant ε > 0 such that, if for any w ∈ FN the translation lengths on T+ and on T− are both smaller than ε (or equal to 0 on one of the two), then w ﬁxes a point in T+ × T− . Finally, Vincent Guirardel has investigated in a much more general context group actions on the cartesian product of two R-trees in [9]. In particular his §9 contains results relevant to this paper.

References [1] M. Bestvina and M. Handel, Train tracks for automorphisms of the free group, Annals of Math. 135, pp. 1–51 (1992) [2] M. Bestvina and M. Feighn, Outer limits, preprint 1992 [3] M. Bestvina, M. Feighn and M. Handel, Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal. 7, pp. 215–244 (1998)

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[4] M. Bestvina, M. Feighn and M. Handel, Erratum to Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal. 7, pp. 1143 (1998) [5] P. Brinkmann, Hyperbolic automorphisms of free groups, Geom. Funct. Anal. 10, pp. 1071–1089 (2000) [6] M. Culler, G. Levitt, P. Shalen, unpublished manuscript [7] M. Feighn and M. Handel, Mapping tori of free group automorphisms are coherent, Ann. of Math. 149, pp. 1061–1077 (1999) [8] D. Gaboriau, A. J¨ ager, G. Levitt and M. Lustig, An index for counting ﬁxed points of automorphisms of free groups, Duke Math. J. 93, pp. 425–452 (1998) [9] V. Guirardel, Coeur et nombre d’intersection pour les actions de groupes sur les arbres, Annales de l’E.N.S. 38, pp. 847–888 (2005) [English version: Core and intersection number for group actions on trees, arXiv math. Gr 0407206] [10] G. Levitt and M. Lustig, Automorphisms of free groups have asymtotically periodic dynamics, to appear in J. reine u. angew. Math. (arXiv math. GR 0407437) [11] M. Lustig, Automorphismen von freien Gruppen, Habilitationsschrift 1992, RuhrUniversit¨ at Bochum [12] M. Lustig, Discrete actions on the product of two non-simplicial R-trees, preprint 1994 [13] M. Lustig, Conjugacy and centralizers for iwip automorphisms of free groups, Geometric Group Theory, Trends in Mathematics, pp. 197–224, Birkh¨auser, Basel 2007 [14] M. Lustig et al., Seven steps to happiness, preliminary preprint 2008 Martin Lustig Math´ematiques (LATP) Universit´e Paul C´ezanne – Aix Marseille III av. Escadrille Normandie-Ni´emen F-13397 Marseille 20, France e-mail: [email protected]

Combinatorial and Geometric Group Theory Trends in Mathematics, 251–260 c 2010 Springer Basel AG

Mather Invariants in Groups of Piecewise-linear Homeomorphisms Francesco Matucci Abstract. We describe the relation between two characterizations of conjugacy in groups of piecewise-linear homeomorphisms, discovered by Brin and Squier in [3] and Kassabov and Matucci in [6]. Thanks to the interplay between the techniques, we produce a simpliﬁed point of view of conjugacy that allows us to easily recover centralizers and lends itself to generalization. Mathematics Subject Classiﬁcation (2000). 20E45; 37E05; 37E10. Keywords. Piecewise-linear homeomorphism groups, conjugacy invariant.

1. Introduction We denote by PL+ (I) the group of orientation-preserving piecewise-linear homeomorphisms of the unit interval I = [0, 1] with ﬁnitely many breakpoints. We will treat only the case of PL+ (I) even if all the results can be adapted to certain subgroups of PL+ (I) of homeomorphisms with certain requirements on the breakpoints and the slopes (for example, Thompson’s group F and the Thompson-Stein groups PLS,G (I) introduced in the works of Stein [8] and Bieri-Strebel [2]). In particular, it is suﬃcient to restrict our study to functions that do not intersect the diagonal, except for the points 0 and 1 (see Section 2 for the motivation). In their work [3] Brin and Squier deﬁne an invariant under conjugacy for maps of PL+ (I) that do not intersect the diagonal. Their description is based on similar earlier work by Mather [7] for diﬀeomorphisms of the unit interval and allows the classiﬁcation of centralizers and the detection of roots of elements. These techniques were originally introduced as an attempt to solve the conjugacy problem in Thompson’s group F (which was then proved to be solvable by Guba and Sapir in [5]). Later on this approach was reﬁned by Gill and Short in [4] and Belk and Matucci [1] to give another proof of the solution to the conjugacy problem The author gratefully acknowledges the Centre de Recerca Matem` atica (CRM) and its staﬀ for the support received during the development of this work.

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in Thompson’s group F . On the other hand, Kassabov and Matucci showed a solution to the simultaneous conjugacy problem in [6] by producing an algorithm to build all conjugators, if they exist. Similarly, these techniques can be used to obtain centralizers and roots as a byproduct. The aim of this note is to show the connection between the techniques in [3] and [6] to characterize conjugacy in groups of piecewise-linear homeomorphisms. By deﬁning a modiﬁed version of Brin and Squier’s invariant and using a mixture of those points of view it is possible to produce a short proof of the description of conjugacy and centralizers in PL+ (I). In particular, the interplay between these two points of view lends itself to generalizations giving a tool to study larger class of groups of piecewise-linear homeomorphisms. This paper is organized as follows. In Section 2 we give a short account of a key algorithm in [6] (the stair algorithm) to build a particular conjugator g for two elements y, z ∈ PL+ (I). In Section 3 we deﬁne a conjugacy invariant (called Mather invariant ) that essentially encodes the characterization of conjugacy in [3] for PL+ (I). In Section 4 we show to use the stair algorithm to simplify the proof of the characterization of conjugacy of [3] using Mather invariants. In turn, in Section 5 we will show how Mather invariants allow us to shorten the arguments in [6] to classify centralizers of elements. We ﬁnish by brieﬂy describing possible extensions of these tools.

2. The stair algorithm for functions in PL< + (I) In this section we will discuss how to ﬁnd a special conjugator g ∈ PL+ (I) for two functions y, z ∈ PL+ (I), if it exists. The idea will be to assume that such a conjugator g exists and obtain conditions that g must satisfy. Deﬁnition 2.1. We denote by PL< + (I) the subset of PL+ (I) of all functions that lie below the diagonal, that is the maps z ∈ PL+ (I) such that f (t) < t for all t ∈ (0, 1). Similarly, we deﬁne the subset PL> + (I) of functions that lie above the diagonal. A function z ∈ PL+ (I) is deﬁned to be a one-bump function if either > z ∈ PL< + (I) or z ∈ PL+ (I). We will restrict to study conjugacy for one-bump functions. The reason for this assumption is easily explained: if two functions y, z ∈ PL+ (I) are conjugate through g, then g −1 (∂Fix(y)) = ∂Fix(g −1 yg) = ∂Fix(z); since the boundary of the set of ﬁxed points of either y or z is ﬁnite, the ﬁrst step to verify conjugacy is to check if ∂Fix(y) and ∂Fix(z) have the same size. If this is the case, we can always build a map h ∈ PL+ (I) such that h−1 (∂Fix(y)) = ∂Fix(z), hence we reduce to check if h−1 yh and z, which share the same boundary of the ﬁxed set, are conjugate; this is true if, for any two consecutive points ti , ti+1 ∈ ∂Fix(z), we can ﬁnd a conjugator gi ∈ PL+ ([ti , ti+1 ]) for the restrictions of h−1 yh and z to [ti , ti+1 ], which are either identity maps or one-bump functions. By restricting the study of conjugacy to the intervals [ti , ti+1 ], we derive our assumption on the maps.

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If z ∈ PL+ (I), we deﬁne initial slope and ﬁnal slope, respectively, to be the numbers z (0) and z (1). It is clear that if two one-bump functions y and z are conjugate, their initial and ﬁnal slope are the same. A more interesting fact is that a conjugator has to be linear in certain boxes around 0 and 1. This fact, together with the ability to identify the two functions step by step, allows us to build a conjugator. Lemma 2.2 (Kassabov and Matucci, [6]). Suppose y, z ∈ PL< + (I). 1. (initial box) Let g ∈ PL+ (I) be such that g −1 yg = z. Assume y(t) = z(t) = ct for t ∈ [0, α] and c < 1. Then the graph of g is linear inside the box [0, α] × [0, α]. A similar statement is true for a “ﬁnal box”. 2. (identiﬁcation trick) Let α ∈ (0, 1) be such that y(t) = z(t) for t ∈ [0, α]. Then there exists a g ∈ PL+ (I) such that z(t) = g −1 yg(t) for t ∈ [0, z −1 (α)] and g(t) = t in [0, α]. The element g is uniquely deﬁned up to the point z −1 (α). 3. (uniqueness of conjugators) For any positive real number q there exists at most one g ∈ PL+ (I) such that g −1 yg = z and g (0) = q. 4. (conjugator for powers) Let g ∈ PL+ (I) and n ∈ N. Then g −1yg = z if and only if g −1 y n g = z n . Proof. The proof of (1) is straightforward. To prove (2) we observe that, if such a g exists then, for t ∈ [0, z −1(α)] y(g(t)) = g(z(t)) = z(t) since z(t) ≤ α. Thus g(t) = y −1 z(t) for t ∈ [0, z −1(α)]. To prove that such a g exists, deﬁne t t ∈ [0, α] g(t) := −1 y z(t) t ∈ [α, z −1 (α)] and extend it to I as a line from the point (z −1 (α), y −1 (α)) to (1, 1). To prove (3), assume that there exist two conjugators g1 , g2 with initial slope q. Since g1−1 yg1 = g2−1 yg2 we have that g := g1 g2−1 centralizes y and it has initial slope 1. Assume, by contradiction, that g is the identity on [0, α] for some α, but g (α+ ) = 1. Since we have y(g(t)) = g(y(t)) = y(t) for t ∈ [α, y −1 (α)], this implies that g(t) = y −1 y(t) = t on [α, y −1 (α)], which is a contradiction. To prove the last statement we observe that if f := g −1 y n g = z n , then f is centralized by both g −1 yg and z. Since g −1 yg and z have the same initial slope, then by (3) we have g−1 yg = z. Part (1) of the previous lemma tells us that any given conjugator g must be linear in two suitable boxes [0, α]2 and [β, 1]2 , hence if we are given a point (p, g(p)) in any of those boxes (say the ﬁnal one), we can draw the longest segment contained in [β, 1]2 passing through (p, g(p)) and (1, 1) and obtain the map g in that box. We are now going to build a candidate conjugator with a given initial slope.

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2 Theorem 2.3 (Stair Algorithm, [6]). Let y, z ∈ PL< + (I), let [0, α] be the initial linearity box and let 0 < q < 1 be a real number. There is an N ∈ N such that the unique candidate conjugator with initial slope q is given by

g(t) = y −N g0 z N (t)

∀t ∈ [0, z −N (α)]

and linear otherwise, where g0 is any map in PL+ (I) which is linear in the initial box and such that g0 (0) = q. By “unique candidate conjugator” we mean a function g such that, if there exists a conjugator between y and z with initial slope q, then it must be equal to g. Hence we can test our candidate conjugator to verify if it is indeed a conjugator. Proof. Let [β, 1]2 be the ﬁnal box and N an integer big enough so that min{z −N (α), y −N (qα)} > β. We will build a candidate conjugator g between y N and z N (if it exists) as a product of two functions g0 and g1 . We note that the linearity boxes for y N and z N are still given by [0, α]2 and [β, 1]2 . By Lemma 2.2(1) g has to be linear on [0, α] and so we deﬁne an “approximate conjugator” g0 by: g0 (t) := qt

t ∈ [0, α]

and extend it to the whole I as a line through (1, 1). We then deﬁne y1 := g0−1 yg0 and look for a conjugator g1 of y1N and z N , noticing that y1N and z N coincide on [0, α]. By the proof of Lemma 2.2(3), we deﬁne t t ∈ [0, α] g1 (t) := −N N y1 z (t) t ∈ [α, z −N (α)] and extend it to I as a line through (1, 1) so that g1−1 y1N g1 = z N on [0, z −N (α)]. Finally, build a function g such that g(t) := g0 g1 (t) for t ∈ [0, z −N (α)] and extend it to I as a line through (1, 1) on [z −N (α), 1]. The map g is inside the ﬁnal box at t = z −N (α) > β, in fact g(z −N (α)) = g0 g0−1y −N g0 (α) = y −N (qα) > β. We observe that, by construction, g is a conjugator for y N and z N on [0, z −N (α)], that is g = g0 g1 = y −N g0 g1 z N on [0, z −N (α)]. Therefore g(t) = y −N g0 g1 z N (t) = y −N g0 z N (t)

∀t ∈ [0, z −N (α)]

since g1 z N (t) = z N (t) for t ∈ [0, z −N (α)]. By parts (1) and (3) of Lemma 2.2, if there is a conjugator for y N and z N with initial slope q, it must be equal to g. So we just check if g conjugates y N to z N . Moreover, Lemma 2.2(4) tells us that g is a conjugator for y N and z N if and only it is for y and z and so we are done. We remark that this proof does not depend on the choice of g0 . The only requirements on g0 are that it must be linear in the initial box and g0 (0) = q.

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3. Mather invariants for functions in PL> + (I) In this section we will give an alternate description of Brin and Squier’s conjugacy invariant in [3]. This reformulation was also used by Belk and Matucci in [1] to characterize conjugacy in Thompson’s group F : however, their proof relies on special kinds of diagrams peculiar to F and cannot be generalized to other groups of homeomorphisms. Roughly speaking, the Mather invariant of a map z ∈ PL> + (I) is deﬁned by taking a power of z large enough so that points very close to 0 get mapped to points very close to 1. We will now deﬁne it precisely. Consider a one-bump function z ∈ PL> + (I), with initial slope m0 and ﬁnal slope m1 . In a neighborhood of zero, z acts as multiplication by m0 : for any suﬃciently small t > 0 and suﬃciently small powers of z, we have z(t) = m0 t, z 2 (t) = m20 t, z 3 (t) = m30 t, . . ., that is the interval [t, m0 t] is a “fundamental domain” for the action of z:

If we make the identiﬁcation t ∼ m0 t in the interval (0, ), for a suﬃciently small > 0, we obtain a circle C0 , with natural projection map p0 : (0, ) → C0 . Similarly, if we identify (1−t) ∼ (1−m1 t) on the interval (1−δ, 1), for a suﬃciently small δ > 0, we obtain a circle C1 , with natural projection map p1 : (1−δ, 1) → C1 . Let > 0 be small enough so that ( , ) surjects onto C0 : if N is suﬃciently large, then z N will take ( , ) and map it to the interval (1 − δ, 1). This induces a map z ∞ : C0 → C1 , making the following diagram commute: zN ( , ) .................................................. (1 − δ, 1) p

... ... ... ... ... 0 ..... ... . ......... ..

C0

... ... ... ... ... ... ... ... ........ ..

p1

z∞

....... ....... ....... ...........

C1

∞

The map z deﬁned above is called the Mather invariant for z. We note that z ∞ does not depend on the speciﬁc value of N chosen. Any map z m , for m ≥ N , induces the same map z ∞ . This is because z “acts as the identity on C1 ”: we can write z m (t) as z m−N (z N (t)), with z N (t) ∈ (1 − δ, 1) and so, by deﬁnition of ∼, we have z m (t) ∼ z N (t). If k > 0, then the map t → kt on (0, ) induces a “rotation” rotk of C0 . In particular, if we use the coordinate θ = log t on C0 , then rotk (θ) = θ + log k so rotk is an actual rotation. In the next section we will give a characterization of conjugacy for one-bump functions by means of Mather invariants.

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4. Equivalence of the two points of view In this section we will show the relation between the stair algorithm and the deﬁnition of Mather invariant. This will provide an alternative proof of Brin and Squier’s conjugacy invariant. Theorem 4.1 (Brin and Squier, [3]). Let y, z ∈ PL> + (I) be one-bump functions with y (0) = z (0) and y (1) = z (1), and let y ∞ , z ∞ : C0 → C1 be the corresponding Mather invariants. Then y and z are conjugate if and only if y ∞ and z ∞ diﬀer by rotations of the domain and range circles: C0

y∞

............................................

C1

... ... ... ... . k ...... . ......... ..

... ... ... ... ... ... .. ......... ..

rot

C0

rot

............................................

z∞

C1

Proof. Since y (0) = z (0) we can pick the fundamental domain for y and z around 0 to be the same. Similarly, we can do it around 1 and so it makes sense to talk about rotations for C0 and C1 . We stress that the Mather invariants y ∞ and z ∞ that we now use depend on the choice of the fundamental domains around 0 and 1 to talk about well-deﬁned compositions. We assume z = g −1 yg for some g ∈ PL+ (I) and follow the notation of the previous section, taking , , δ > 0 small enough and N large enough. Then z N = g −1 y N g and the following diagram commutes, where k = g (0) and = g (1): yN ( , ) ................................................................... (1 − δ, 1) g ................. ........ g ................. ........

( , ) ... ... ... ... . ..... ... ... ... 0 ...... ... ... ... ... ... . ......... ..

... ..

.. ... ....

zN ................................................................... .

... ... ... ... ... ... ... ......... ..

(1 − δ, 1)

p0

p

......... ... . .... ... . . .... ....

C0

.. ... ....

y∞

.......................................

rotk

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ......... .

p1 .........................

...... ..... .... .... . . . .... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ......... ..

p1

C1

rot

z To show the converse, choose g0 ∈ PL+ (I) that is linear in the initial box and such that g0 (0) = k and deﬁne the map g to be the following pointwise limit C0

........................................................................................

∞

C1

g(t) := lim y n g0 z −n (t). n→∞

By the Stair Algorithm (Theorem 2.3) it is clear that g conjugates y −1 to z −1 (and hence y to z). It remains to show that g ∈ PL+ (I). By construction, g has only ﬁnitely many breakpoints on the interval [0, 1 − δ] for a δ > 0 small enough. Since

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g conjugates y and z, then g induces a well-deﬁned map gind : C1 → C1 (given by p1 gp−1 1 ) and we can build a diagram similar to the one of “only if” part of this theorem yN ( , ) ................................................................... (1 − δ, 1) g ................ ........ g ................. ........ ... ..

.. ... ....

( , )

zN ..................................................................

.. ... ....

(1 − δ, 1)

... ... ... ... ... ... ... ... ... ... ... ... 0 ... ... ... . . . ........ ... ∞ 0 ...... ... ....................................... ... 0 ... ... ......... . ... .... . . ... ... ......... .. ...... . . . k ........................................................................................

p

p

C

y

rot

C0

... ... ... ... ... ... ... ... ... ... ... ... ... ... ......... ..

p1

... ... ... ... ... ... ... ... ... 1 ... ... ... ......................... ... 1 ... ... ......... ... .... . .. ... . . . ........ .. . ...... ind ..

p

C

g

C1

z∞ for suitable , , δ > 0 small enough and an integer N big enough. By hypothesis the Mather invariants diﬀer by rotations of the domain and range circles, therefore we have rot z ∞ = y ∞ rotk = gind z ∞

and so, by cancellation, gind is a rotation by . To prove that g ∈ PL+ (I) we show that g is linear around 1 in the following claim: Claim: If g : I → I is a continuous map and p1 is a projection of a neighbourhood of 1 to C1 such that p1 gp−1 1 is a well-deﬁned map from C1 to C1 and it is a rotation of C1 , then g is linear on (1 − δ, 1] for a δ > 0 small enough. Proof of the claim. Let δ > 0 be small enough so that (1 − δ, 1] is contained in the domain of p1 and let t ∈ (1 − δ, 1]. Following the notation from Section 3, since p1 gp−1 1 is a rotation by , we have g(t) = g(1 − (1 − t)) = 1 − mr1 (1 − t). for some integer r. Thus, for a λ > 0 close enough to 1, we have g(1 − λ(1 − t)) = 1 − mr1 λ(1 − t) = 1 − λ(1 − g(t)). By the previous equation, the function h(t) :=

1 − g(t) 1−t

satisﬁes h(t) = h(1 − λ(1 − t)) for λ > 0 close enough to 1, hence h is locally constant on (1 − δ, 1] and therefore it is constant. Since h is constant, the map g is then linear around 1. Remark 4.2. We have slightly abused the notation in the two cube diagrams of the previous proof: to simplify the exposition, we have not been careful in choosing

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the range sets for g that still surject onto C0 and C1 (although it can be made precise). Remark 4.3. The previous proof shows that two functions y, z are conjugate if and only if the Stair Algorithm builds a linear map in the ﬁnal linearity box and this happens if and only if the two Mather invariants diﬀer by rotations of the domain and the range circles. The Mather invariant thus gives the “obstruction” to ﬁnishing the Stair Algorithm at 1. Remark 4.4. We stress that the deﬁnition of Mather invariant and the construction of the stair algorithm do not really depend upon the set of breakpoints and slopes of the maps y and z. With little work, the two constructions and their equivalence can be extended to Thompson-Stein groups (see also [6]).

5. Applications: centralizers and generalizations Given a map f : S 1 → S 1 , a lift of f is a map F : R → R such that F (t + 1) = F (t) + 1 for all t ∈ R and F induces f when passing the domain and the range to quotients via the relation α ∼ α + 1. Given a lift, we talk about a maximal V interval to refer to an interval [a, b] such that F is linear with slope V on [a, b] and a, b are breakpoints for F . We will give a short proof of the following well-known result. Theorem 5.1. Let z ∈ PL> + (I). Then the centralizer subgroup CPL+ (I) (z) = {g ∈ PL+ (I) | gz = zg} is isomorphic to the inﬁnite cyclic group. Proof. Deﬁne the following group homomorphism: ϕz : CPL+ (I) (z) −→ (R, +) g −→ log g (0). Lemma 2.2(3) implies that ϕz is injective. By Theorem 4.1 any function g centralizing z induces two rotations rot , rotk such that rot z ∞ = z ∞ rotk where k = g (0) and = g (1). Observe that R (t) = t + log and Rk (t) = t + log k are lifts of the two rotations rot , rotk . Choose a lift Z : R → R of z ∞ . The previous equality implies: Z(t) + log = R (Z(t)) = Z(Rk (t)) = Z(t + log k) which means that the graph of Z can be shifted “diagonally” onto itself. The map Z is piecewise-linear and, for any positive number r, has ﬁnitely many breakpoints on the interval [−r, r]. Hence Z has only ﬁnitely many maximal Z (0)-intervals that are contained in [−r, r] and so there is only a discrete set of shifts (that is, values of log k = ϕz (g)) which maps the graph of Z onto itself, unless Z is a line. To see that this is not the case, we show that z ∞ has breakpoints. Let N be a power large enough so that a fundamental domain near 0 is sent near 1 so

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that z N induces z ∞ , then either z −N or z −2N has a breakpoint in the ﬁnal box [β, 1]2 (this implies immediately that z ∞ must have breakpoints). If they were both linear, by applying the chain rule on z −2N = z −N ◦ z −N ﬁrst at β and then at 1, one sees that the slope z −2N on [β, 1] is simultaneously equal to the product of the slopes z (0)−N (z −N ) (β + ) and z (1)−N (z −N ) (1) and this is impossible since (z −N ) (β + ) = (z −N ) (1), but z (0) < z (1). We have thus proved that the image of ϕz must be a discrete subgroup of (R, +) and so, by a standard fact, it is isomorphic to Z. The Mather invariant approach is also interesting because it lends itself to generalizations. Let PLdis (R) the group of all orientation-preserving piecewiselinear homeomorphisms of the real line with a discrete set of breakpoints and let EP be the subgroup of PLdis (R) of the functions that are “eventually periodic at inﬁnity”, that is functions f ∈ PLdis (R) such that there exist numbers Lf , Rf so that f (t − 1) = f (t) − 1 for t < Lf and f (t + 1) = f (t) + 1 for t > Rf . It is easy to deﬁne the subset EP> and Mather invariant for functions in EP> : we just mod out the intervals (−∞, Lf ) and (Rf , ∞) by the relation t ∼ f (t) and then take a power of f high enough so that (f −1 (Lf ), Lf ) gets carried to a subset of (Rf , ∞). Similarly, one can partially extend the stair algorithm to build conjugators. It is thus interesting to see how much of these techniques can be extended to overgroups containing PL+ (I) to compute centralizers and, possibly, to study the conjugacy problem. Acknowledgment The author would like to thank Ken Brown, Jos´e Burillo, Martin Kassabov and an anonymous referee for helpful comments that improved the presentation of this paper.

References [1] J.M. Belk and F. Matucci. Dynamics in Thompson’s group F . Submitted. arXiv:math.GR/0710.3633v1. [2] R. Bieri and R. Strebel. On groups of PL-homeomorphisms of the real line. notes, 1985. Math. Sem. der Univ. Frankfurt. [3] Matthew G. Brin and Craig C. Squier. Presentations, conjugacy, roots, and centralizers in groups of piecewise linear homeomorphisms of the real line. Comm. Algebra, 29(10):4557–4596, 2001. [4] N. Gill and I. Short. Conjugacy, roots, and centralizers in Thompson’s group F . Preprint. arXiv:math.GR/0709.1987v2 . [5] Victor Guba and Mark Sapir. Diagram groups. Mem. Amer. Math. Soc., 130(620): viii + 117, 1997. [6] M. Kassabov and F. Matucci. The simultaneous conjugacy problem in groups of piecewise linear functions. preprint. arXiv:math.GR/0607167v2.

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[7] John N. Mather. Commutators of diﬀeomorphisms. Comment. Math. Helv., 49:512– 528, 1974. [8] Melanie Stein. Groups of piecewise linear homeomorphisms. Trans. Amer. Math. Soc., 332(2):477–514, 1992. Francesco Matucci Centre de Recerca Matem` atica Apartat 50 E-08193 Bellaterra, Barcelona, Spain e-mail: [email protected]

Combinatorial and Geometric Group Theory Trends in Mathematics, 261–279 c 2010 Springer Basel AG

Algebraic Geometry over the Additive Monoid of Natural Numbers: Systems of Coeﬃcient Free Equations Pavel V. Morar and Artem N. Shevlyakov Abstract. In the paper we consider homogeneous systems of linear equations and classify coordinate monoids over the additive monoid of natural numbers which are deﬁned by such systems. Further, we apply our results to the wide class of commutative monoids. Mathematics Subject Classiﬁcation (2000). 20M14. Keywords. Universal algebraic geometry, natural numbers, equations, coordinate monoids.

1. Introduction In the paper [1] by E. Daniyarova, A.G. Myasnikov, and V. Remeslennikov the following uniﬁcation theorems (Theorem A and B in [1]) were formulated and proved. Theorem 1.1. [No coeﬃcients, [1]] Let B be an equationally Noetherian algebra in a functional language L. Then for a ﬁnitely generated algebra C of L the following conditions are equivalent: 1. Th∀ (B) ⊆ Th∀ (C), i.e., C ∈ ucl(B); 2. Th∃ (B) ⊇ Th∃ (C); 3. C embeds into an ultrapower of B; 4. C is discriminated by B; 5. C is a limit algebra over B; 6. C is deﬁned by a complete atomic type in the theory Th∀ (B) in L; 7. C is the coordinate algebra of an irreducible algebraic set over B deﬁned by a system of coeﬃcient-free equations. Theorem 1.2. [With coeﬃcients, [1]] Let A be an algebra in a functional language LA and B an A-equationally Noetherian A-algebra. Then for a ﬁnitely generated A-algebra C the following conditions are equivalent:

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Th∀,A (B) ⊆ Th∀,A (C), i.e., C ∈ uclA (B); Th∃,A (B) ⊇ Th∃,A (C); C A-embeds into an ultrapower of B; C is A-discriminated by B; C is a limit algebra over B; C is an algebra deﬁned by a complete atomic type in the theory Th∀,A (B) in LA ; 7. C is the coordinate algebra of an irreducible algebraic set over B deﬁned by a system of equations with coeﬃcients in A.

1. 2. 3. 4. 5. 6.

The equivalence of the conditions from these theorems holds for every algebra B. If we deal with a concrete algebra, its coordinate algebra has additional properties. In our paper we consider positive commutative monoids with cancellation (a monoid is called positive if every sum of non-zero elements is non-zero) and we prove several theorems about the classiﬁcation of coordinate monoids in the case of systems of equations with no coeﬃcients. Our theorems supplement and specialise Theorem 1.1 to a particular case. Algebraic geometry over torsion-free abelian groups is quite simple and was investigated in [2], where the following facts were proved. • Every coordinate group over a torsion-free abelian group A of ﬁnite rank is isomorphic to a free abelian group of ﬁnite rank. • Each algebraic set over a torsion-free abelian group A of ﬁnite rank is irreducible. In this paper we show that algebraic geometry over commutative positive monoids with cancellation is more complicated than over torsion-free abelian groups. Here are the main results. Theorem A. Let N be a commutative positive monoid with cancellation. A ﬁnitely generated monoid M is an irreducible coordinate monoid over N for a system of equations with no coeﬃcients iﬀ M is a commutative positive monoid with cancellation. Theorem B. Let N be a positive commutative monoid with cancellation. Then every coordinate monoid over N for a system of coeﬃcient-free equations is irreducible. Hence every algebraic set over N is irreducible. According to the following theorem, proved in our paper, it is only necessary to prove Theorems A and B in the case N = N, where N is the additive monoid of natural numbers. Theorem C. Let M be a commutative positive monoid with cancellation and S be a system of equations with no coeﬃcients. Then the coordinate algebra of S over M is equal to the coordinate algebra of S over N, where N is the additive monoid of natural numbers.

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In the ﬁnal section we develop the dimension theory of algebraic sets over the additive monoid of natural numbers N and oﬀer a method of calculation of the dimension of an algebraic set. Theorem D. Let Y be an algebraic set over the additive monoid of natural numbers N for a system of equations with no coeﬃcients. Then the dimension of Y is equal to the dimension of the subspace ρ(Y ) of the Euclidian space Rn , generated by Y . In conclusion the authors would like to thank Professor V. Remeslennikov for his support, attention and constructive criticism.

2. A-monoids Let A be a monoid. A monoid B is said to be an A-monoid if B contains a submonoid S such that S and A are isomorphic. More precisely, an A-monoid is a pair (B, λ), where λ : A → B is embedding. Suppose C is a submonoid of B and C ⊇ λ(A); then a pair (C, λ) is called an A-submonoid of (B, λ). We say that an A-monoid B is ﬁnitely generated over A if B is generated by A and a ﬁnite set C. Let B1 and B2 be A-monoids, λ1 : A → B1 and λ2 : A → B2 be embeddings. A homomorphism ϕ : B1 → B2 is called an A-homomorphism if ϕ(λ1 (a)) = λ2 (a) for each a ∈ A. Denote by HomA (B1 , B2 ) the set of all A-homomorphisms mapping B1 to B2 . Suppose B, C are A-monoids. We say that C is A-separated by B if for each pair of distinct elements c1 , c2 ∈ C there exists an A-homomorphism ϕ : C → B such that ϕ(c1 ) = ϕ(c2 ). We say that C is A-discriminated by B if for each natural number k and for each set c1 , . . . , ck consisting of k pairwise distinct elements of C, there exists an A-homomorphism ϕ : C → B such that ϕ(ci ) = ϕ(cj ) with i = j. Denote by ResA (B) and DisA (B) the classes of all A-monoids which are Aseparated and A-discriminated by B respectively. 2.1. Logical preliminaries Let L = ◦(2) , 1 be the standard language of monoid theory. L contains the binary functional symbol ◦ and the constant symbol 1 with the standard interpretation. We extend the language L to LA = L ∪ {ca |a ∈ A}. The class of all A-monoids is deﬁned by the following series of LA -axioms: I) The axioms of monoid theory: Ax : ∀x∀y∀z (x ◦ y) ◦ z = x ◦ (y ◦ z); Ax : ∀x x ◦ 1 = 1 ◦ x = x; II) The axioms which describe the existence of a submonoid which is isomorphic to A: • ca = 1 for a = 1; • ca1 ◦a2 = ca1 ◦ ca2 ; • ca1 = ca2 iﬀ a1 = a2 .

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Now we give the deﬁnitions of two special types of LA -formulas. • A universal sentence is a formula ∀x1 . . . ∀xn ϕ(x1 , . . . , xn ), where ϕ is a quantiﬁer-free formula. • A quasi-identity is a universal sentence ∀x1 . . . ∀xn ϕ(x1 , . . . , xn ), where ϕ is of the form ∧m x) = si (¯ x) i=1 ti (¯

→

r1 (¯ x) = r2 (¯ x),

and si , ti , r1 , r2 are LA -terms. Let B be an A-monoid. Below we give the deﬁnitions of two special classes of A-monoids. • The A-universal closure of B is a set of all A-monoids C such that, for each universal LA -formula ψ, if ψ is true in B then C |= ψ. • The A-quasivariety of B is a set of all A-monoids C such that, for each LA -quasi-identity ϕ, if ϕ is true in B then C |= ϕ. Denote by uclA (B) and qvarA (B) the A-universal closure and the A-quasivariety of B. It is easy to show that uclA (B) ⊆ qvarA (B).

3. Introduction to algebraic geometry In this section we give basic deﬁnitions and main theorems of algebraic geometry over monoids. For more details see [1]. 3.1. Systems of equations Suppose X = {x1 , . . . , xn } is a ﬁxed set of variables. Let TLA (X) be the set of LA -terms with variables from the set X. The composition of terms on the set TLA (X) is clearly deﬁned. TLA (X) is said to be an absolutely free algebra with the basis X. An atomic LA -formula t = s, where t, s ∈ TLA (X), is called an equation over A, and a set of equations S = {ti = si | i ∈ I, ti , si ∈ TLA (X)} is called a system of equations over A. Only variables, the symbols ◦, 1 and the constants ca (a ∈ A) can occur in a term t ∈ TLA (X); thus the value t(b1 , . . . , bn ), bi ∈ B is deﬁned for every Amonoid B. Hence we can seek a solution of every equation over A in an arbitrary A-monoid B. The n-dimensional aﬃne space over B is the set B n = {(b1 , . . . , bn ) | bi ∈ B}. A point p = (b1 , . . . , bn ) ∈ B n is said to be a solution of an equation t = s if B |= t = s with the interpretation xi → bi , 1 ≤ i ≤ n. Moreover, p is called a solution of the system of equations S if p is a solution of every equation of S. Denote by VB (S) the set of all solutions of a system S.

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A system S is called incompatible over B if VB (S) = ∅, and otherwise S is compatible. If the monoid A is trivial, i.e., A = {1}; then the language LA is equal to L, and every equation over A is coeﬃcient-free. In this case the algebraic geometry over B is called coeﬃcients free. 3.2. Algebraic sets A set Y ⊆ B n is said to be algebraic over B if there exists a system of equations S such that Y = VB (S). An algebraic set Y is called irreducible if there do not exist algebraic sets Y1 , Y2 such that Y = Y1 ∪ Y2 , Y = Y1 , and Y = Y2 . Otherwise, Y is a reducible set. Suppose Y ⊆ B n and Z ⊆ B m are algebraic sets. A mapping μ : Y → Z is called an LA -mapping if there are terms t1 , . . . , tm ∈ TLA (X) such that μ(b1 , . . . , bn ) = (t1 (b1 , . . . , bn), . . . , tm (b1 , . . . , bn )) ∈ Z. Two algebraic sets Y, Z are called isomorphic if there are two LA -mappings μ : Y → Z, η : Z → Y such that η ◦ μ and μ ◦ η are the identity mappings over Y and Z respectively. Lemma 3.1. Suppose Y, Z are isomorphic sets; then Y is irreducible iﬀ Z is irreducible. Now we deﬁne the category AS A (B) of all algebraic sets over B. The set of objects of AS A (B) consists of all nonempty algebraic sets over B, and the set of morphisms of this category is the set of all LA -mappings of algebraic sets over B. Note that the empty set is algebraic over an A-monoid B whenever A = {1} because ∅ is deﬁned by the system 1 = a, where a ∈ A\{1}. If A = {1}, the empty set is not algebraic since every system S with the variables X = {x1 , . . . , xn } is compatible and has the solution (1, 1, . . . , 1). In other words, we have the following statement. Statement 3.2. ∅ ∈ AS A (B) iﬀ A = {1}. 3.3. Radicals Let Y be an arbitrary set in B n . The radical of Y is the set of all equations over A which are satisﬁed by all points of Y . More formally, RadB (Y ) = {t(¯ x) = s(¯ x) | B |= t(p) = r(p)

∀ p ∈ Y }.

It is clear that the radical RadB (∅) contains all equations over A. The radical of a system S is RadB (S) = RadB (VB (S)). The following proposition describes the properties of radicals. Proposition 3.3. • Suppose Y1 ⊆ Y2 ⊆ B n ; then RadB (Y1 ) ⊇ RadB (Y2 ).

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• Suppose Yi ⊆ B n , i ∈ I; then

RadB Yi = RadB (Yi ). i∈I

i∈I

• Each radical deﬁnes a unique algebraic set, i.e., if Y1 , Y2 ⊆ B n are algebraic sets, we have Y1 = Y2 ⇔ RadB (Y1 ) = RadB (Y2 ). Suppose Y ⊆ B n . Proposition 3.3 states that the radical RadB (Y ) deﬁnes an algebraic set Y¯ = VB (RadB (Y )), which is called the algebraic closure of Y . It is easy to see that the set Y is algebraic whenever Y¯ = Y . 3.4. Coordinate monoids Let ∼ be an equivalence relation on an A-monoid C. The relation ∼ is called congruence if the following condition holds m 21 ◦ m 22 1 ◦ m2 = m

(1)

for all equivalence classes m 21 , m 22 , where m 2i is the equivalence class of an element mi ∈ C. It is clear that each radical determines a congruence over TLA (X). The congruence generated by RadB (Y ) is denoted by ΘRadB (Y ) . Hence for each algebraic set Y there is the factor monoid ΓA (Y ) = TLA (X)/ΘRadB (Y ) . The monoid ΓA (Y ) consists of the equivalence classes of terms from TLA (X), and the product of two classes is deﬁned by formula (1). It is easy to check that for every nonempty algebraic set Y the factor monoid ΓA (Y ) is an A-monoid, and for Y = ∅ we have ΓA (Y ) = {1}. The factor monoid ΓA (Y ) is called the coordinate A-monoid of the algebraic set Y . Let p = (b1 , . . . , bn ) be a point of the aﬃne space B n . There is a mapping ϕp : TLA (X) → B deﬁned by ϕp (t) = t(b1 , . . . , bn ). This mapping has the following properties. Proposition 3.4. Suppose Y ⊆ B n is an algebraic set. • If p ∈ Y is a point, then ϕp ∈ HomA (ΓA (Y ), B). • If ϕ ∈ HomA (ΓA (Y ), B), then there exists a point p ∈ B n such that ϕ = ϕp . Let us deﬁne the category CMA (B). The set of objects of CMA (B) consists of all coordinate A-monoids corresponding to nonempty algebraic sets over B. The set of morphisms of this category consists of all A-homomorphisms ϕ : Γ1 → Γ2 . The following proposition establishes relations between AS A (B) and CMA (B). Proposition 3.5. Two algebraic sets Y1 , Y2 are isomorphic whenever ΓA (Y1 ) and ΓA (Y2 ) are isomorphic. Remark 3.6. Further we write Y ∈ AS A (B) if Y is an algebraic set over B and we write M ∈ CMA (B) if M is a coordinate monoid corresponding to an algebraic set over B.

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We shall say that a coordinate A-monoid ΓA (Y ) is irreducible if Y is irreducible. It follows from Lemma 3.1 and Proposition 3.5 that this deﬁnition is well deﬁned. 3.5. Equationally Noetherian monoids An A-monoid B is said to be A-equationally Noetherian if for every inﬁnite system of equations S that depends on variables x1 , . . . , xn , there exists a ﬁnite system S0 ⊆ S such that VB (S) = VB (S0 ). If a monoid B is A-equationally Noetherian we need only consider ﬁnite systems of equations. Moreover, the following theorem holds. Theorem 3.7. Let B be an A-equationally Noetherian monoid. Then every algebraic set over B can be represented as a ﬁnite union of irreducible algebraic sets Yi (irreducible components). Moreover, if Yi Yj for i = j, then this decomposition is unique up to a permutation of components. The next theorem was proved in [2] for groups, but its proof can be easily extended to arbitrary algebraic systems of a language without relations. We formulate this theorem for monoids. Theorem 3.8. Let B be an A-equationally Noetherian monoid. Then for an arbitrary A-monoid C the following conditions are equivalent: 1. C is a coordinate monoid of an algebraic set over B (C ∈ CMA (B)); 2. C is A-separated by B (C ∈ ResA (B)); 3. C ∈ qvarA (B); 4. C ∈ SP(B), where S, P are the operators of taking A-submonoids and direct products respectively. Remark 3.9. According to Theorem 3.8, each monoid from quasivariety qvarA (B) can be obtained from B as the result of a composition of the operators S, P. Obviously, S, P preserve the Noetherian property, and so if B is A-equationally Noetherian, then M is A-equationally Noetherian too. The main problem of algebraic geometry over an A-monoid B is to classify • algebraic sets of the category AS A (B); • radicals of algebraic sets and • coordinate A-monoids of the category CMA (B). Propositions 3.3 and 3.5 establish the equivalence of these three approaches. In our paper we follow the third approach and Theorems 1.1 and 3.8 give us a powerful tool for the classiﬁcation of coordinate monoids.

4. Commutative monoids with cancellation In this section we begin to study algebraic geometry over the additive monoid of natural numbers and give some properties of commutative monoids with cancellation.

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In the sequel we reserve the notation A for a ﬁxed submonoid of N. The following proposition plays an important role in the paper. Proposition 4.1. The monoid N is A-equationally Noetherian. Proof. Let R be the additive monoid of real numbers and let P be a submonoid of R. It follows from well-known theorems of linear algebra that R is P -equationally Noetherian, and the proof of this proposition immediately follows from Remark 3.9 and the inclusion N ⊂ R. Theorem 3.8 states that all coordinate A-monoids of CMA (N) belong to quasivariety qvarA (N). Hence all quasi-identities that are true in N must be true in every A-monoid of CMA (N). In particular, all monoids of CMA (N) must be commutative because the axiom of commutativity Axcomm : ∀x∀y x ◦ y = y ◦ x ∼ ∀x∀y (1 = 1) → x ◦ y = y ◦ x

(2)

is a quasi-identity. The commutativity of all coordinate monoids allows us to use the traditional symbols +, 0 instead of ◦, 1, and further we use the language +, 0 as L. Therefore, each LA -term is logically equivalent to an LA -term of the form (3) αi xi + a, αi ∈ N, a ∈ A. Remark 4.2. According to formula (3), we can consider elements of the absolutely free algebra TLA (X) as elements of the free commutative monoid generated by A and X = {x1 , . . . , xn }. A commutative monoid M is called a monoid with cancellation if the quasiidentity (4) Axcanc : ∀x∀y∀z x + z = y + z → x = y holds in M . Theorem 4.3 ([5]). Suppose M is a commutative monoid. Then M embeds into the additive group whenever M is a monoid with cancellation. All coordinate A-monoids of algebraic sets over N are monoids with cancellation because N |= Axcanc . Therefore, each equation t = s over A can be written in the form γi xi + a = γj xj + a , (5) i∈I

j∈J

where a, a ∈ A and I ∩ J = ∅. Remark 4.4. It is easy to see that there may be equations over A that can not be transformed to the form γi xi + a = γj xj , i∈I

j∈J

where one side of the equation does not have constant term.

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5. Coeﬃcient free algebraic geometry over N In this section we classify coordinate monoids of the category CM0 (N). Deﬁnition 5.1. A monoid is called positive if for every pair of nontrivial elements m1 , m2 ∈ M the sum m1 + m2 is not equal to 0. We explain the motivation for the term “positive” in Subsection 5.2. Remark 5.2. It is easy to prove that a monoid M is positive whenever the set M \{0} is a semigroup. The quasi-identity of the language L Axpos : ∀x∀y x + y = 0 → x = 0,

(6)

describes the property of positiveness. The following important theorems give us a full description of coordinate monoids of the category CM0 (N). Theorem A. A ﬁnitely generated monoid M is an irreducible coordinate monoid of the category CM0 (N) iﬀ M is a commutative positive monoid with cancellation (M |= Axcomm , Axcanc , Axpos ). We prove Theorem A in Subection 5.3. Theorem B. Every coordinate monoid of the category CM0 (N) is irreducible. Hence every algebraic set of the category AS 0 (N) is also irreducible. Proof. Theorem B is a simple corollary of Theorem A and Theorem 3.8. Indeed, suppose M ∈ CM0 (N). It follows from Theorem 3.8 that M ∈ qvar0 (N). The axioms Axcomm , Axcanc , Axpos are deﬁned as the quasi-identities (2), (4), (6), so we see that M |= Axcomm , Axcanc , Axpos . It follows from Theorem A that the monoid M is irreducible. The next corollary follows from Theorems 1.1, 3.8, A, and B. Corollary 5.3. The universal class qvar0 (N) is equal to the universal class ucl0 (N), and the set of formulas of the language L = +, 0 1. ∀x∀y∀z (x + y) + z = x + (y + z); 2. ∀x x + 0 = 0 + x = x; 3. ∀x∀y x + y = y + x 4. ∀x∀y∀z x + z = y + z → x = y; 5. ∀x∀y x + y = 0 → x = 0. is a complete set of axioms of these classes. In the following subsection we study properties of ﬁnitely generated commutative positive monoids with cancellation. These properties are necessary for the proof of Theorem A (Theorem C is proved as a corollary of Theorem A in Subsection 6).

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5.1. Properties of ﬁnitely generated commutative positive monoids with cancellation Let us study the properties of ﬁnitely generated monoids that satisfy the axioms Axcomm , Axcanc , Axpos . The next proposition holds for these monoids. Proposition 5.4. Suppose M is a ﬁnitely generated commutative positive monoid with cancellation. Then there exists a natural number n such that M is embedded in the additive group Zn , and also this group can be embedded in the Euclidian space Rn so that vectors corresponding to elements of the monoid M have integer coordinates. Proof. According to Theorem 4.3, there exists an abelian group G and an embedding ϕ : M → G. It is clear that G is ﬁnitely generated. The group G is isomorphic to the direct sum G = Z1 ⊕ · · · ⊕ Zn ⊕ t(G), where Zi Z and t(G) is the torsion part of G. The monoid M is positive, so it is torsion-free and ϕ(M )∩t(G) = ∅. Therefore ϕ is an embedding M → Zn . Recall some well-known facts of theory of convex analysis in the Euclidian spaces. Let C ⊆ Rn be a set. The intersection of all linear varieties of Rn that contain C is called an aﬃne hull of the set C and denoted by aﬀ(C). A relative interior ri(C) of the set C is a subset of C such that every point of ri(C) has a neighborhood O with O ∩ aﬀ(C) ⊆ C. It follows from the deﬁnition that ri(C) is an open set of aﬀ(C). A set K ⊆ Rn is called a cone if for every point x ∈ K and for every positive real number λ we have λx ∈ K. A set C ⊆ Rn is called convex if for all points x1 , x2 ∈ C and for every real number 0 λ 1 the point λx1 + (1 − λ)x2 is in C. The next two theorems of convex analysis give us important properties of convex sets in Rn . Theorem 5.5 ([6]). If a set K ⊆ Rn of vectors is closed under addition and multiplication by positive real numbers, then K is a convex cone. Theorem 5.6 ([6]). Suppose K1 , K2 are nonempty convex cones in Rn and ri(K1 )∩ ri(K2 ) = ∅. Then there exists a nontrivial vector a such that (a, x) ≥ 0 for each x ∈ K1 , (a, x) ≤ 0 for each x ∈ K2 , and there exists a point x0 ∈ K1 ∪ K2 satisfying strict inequalities.

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Let M ⊆ Zn be a ﬁnitely generated monoid with a generating set {m1 , . . ., mk }. Denote k 3 LR+ (M ) = αi mi | αi ∈ R, αi ≥ 0 . i=1

This is the linear hull of generators m1 , . . . , mk in the space Rn such that all the coeﬃcients in the sum are real and nonnegative. It is easy to see that the set LR+ (M ) is closed under addition and multiplication by nonnegative real numbers. Therefore, it follows from Theorem 5.5 that this set is a convex cone. For every monoid M ⊆ Zn we can deﬁne the monoid −M = {−m | m ∈ M } ⊆ Zn . Note that M ∩ −M = {0} if M is positive. The next two lemmas hold for ﬁnitely generated positive submonoids of the group Zn . Lemma 5.7. Suppose M ⊆ Zn is a ﬁnitely generated positive monoid. Then LR+ (M ) ∩ LR+ (−M ) = {0}. Proof. Let {m1 , . . . , mk } be a set of generators of M . Assume the converse, i.e., there exists a nonzero vector y ∈ LR+ (M ) ∩ LR+ (−M ). We have y=

k

αi mi , (αi ∈ R+ ),

i=1

y=

k

βi (−mi ) (βi ∈ R+ ).

i=1

Therefore, k k (αi + βi )mi = γi mi = 0 i=1

i=1

γk , and not all of these numbers equal 0. for nonnegative real numbers γ1 , . . . , Without loss of generality it can be assumed that γ1 , . . . , γp are positive and γ4 γk are equal to 0. p+1 , . . . , Let us consider the system of linear equations Mγ = 0 with respect to the variables γ = (γ1 , . . . , γp ), where the ith column of the matrix M is the vector mi . This system can be rewritten in the form p γi mi = 0. (7) i=1

It is obvious that the vector ( γ1 , . . . , γ p ) with positive coordinates is a solution of the system (7). Since there exists a nontrivial solution, the set of free variables is nonempty.

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Let γ1 , . . . , γl be free variables and γl+1 , . . . , γp be dependent variables. Since all elements of the matrix M are integers and the algorithm of Gaussian elimination uses only arithmetic operations (addition, subtraction, multiplication, division), we can represent the variables γl+1 , . . . , γp as a linear combination of the free variables γ1 , . . . , γl with rational coeﬃcients. Now let us prove that there is a nonzero positive rational solution (r1 , . . . , rp ) of (7). If all coordinates of the vector ( γ1 , . . . , γp ) are rational our proof is ﬁnished. Suppose that some of elements of ( γ1 , . . . , γ p ) are irrational. We can consider the variables γl+1 , . . . , γp as linear functions with the arguments γ1 , . . . , γl . It is obviγ1 , . . . , γl ). Hence ous that γl+1 , . . . , γp are continuous and positive at the point ( there is a neighbourhood Oγ of ( γ1 , . . . , γl ) such that the functions γl+1 , . . . , γp are positive at every point x ∈ Oγ . Let (r1 , . . . , rl ) be an arbitrary point with rational coordinates from Oγ . The values r1 , . . . , rl of the free variables γ1 , . . . , γl determine a positive rational solution (r1 , . . . , rp ) of the system (7). Since (r1 , . . . , rp ) is a solution of (7), we can substitute (r1 , . . . , rp ) instead of the variables (γ1 , . . . , γp ). We obtain a nontrivial linear combination with positive rational coeﬃcients p ri mi = 0. i=1

If we multiply the both sides of the last equation by a suitable number, we obtain a nontrivial linear combination with natural coeﬃcients. We obtain a contradiction because M is a positive monoid. This contradiction proves that LR+ (−M ) LR+ (M ) = {0}. Lemma 5.8. Suppose M ⊆ Zn is a ﬁnitely generated positive monoid. Then there exists a vector a ∈ Rn such that for every nontrivial element m ∈ M the scalar multiplication (a, m) is positive. Proof. Since the lemma obviously holds for the trivial monoid M = {0}, we assume that M = {0}. Let O0 be a neighbourhood of the zero vector in the space Rn . Since LR+ (M ), LR+ (−M ) are convex cones, we have LR+ (M ) ∩ O0 = {0}, LR+ (−M ) ∩ O0 = {0} and LR+ (M ) ∪ LR+ (−M ) ⊂ aﬀ(LR+ (M )) and aﬀ(LR+ (M )) = aﬀ(LR+ (−M )). We obtain O0 ∩ aﬀ(LR+ (M )) LR+ (M ), O0 ∩ aﬀ(LR+ (−M )) LR+ (−M ). From the two previous formulas we obtain 0∈ / ri(LR+ (M )), 0 ∈ / ri(LR+ (−M )).

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Therefore, ri(LR+ (M )) ∩ ri(LR+ (−M )) = ∅. According to Theorem 5.6, there is a vector a ∈ Rn such that (a, x) ≥ 0 for all x ∈ LR+ (M ) and there exists a point p of LR+ (M ) such that (a, p) > 0. Let {m1 , . . . , mk } be a set of generators of M . Since M ⊆ LR+ (M ), we have (a, m) ≥ 0 for all m ∈ M . If the generators m1 , . . . , mk are orthogonal to a, then each vector from LR+ (M ) is also orthogonal to a and there does not exist a vector x ∈ LR+ (M ) such that (a, x) > 0. Therefore, there exists a generator mi such that the scalar multiplication (a, mi ) is positive. Further we prove by induction on the number of generators of M the existence of a vector a ∈ Rn such that (a , x) > 0 for each element x ∈ M . Clearly, if M = m1 we may set a = m1 . Let us prove the step of induction. Suppose M = m1 , . . . , mk ⊆ Zn . Using the a found above, without loss of generality it may be assumed that m1 = mi , so (a, mi ) > 0 and (a, m) ≥ 0 for all m ∈ M . The inductive assumption gives b such than (b, mi ) > 0 for all i ≥ 2. Since the scalar multiplication is a continuous function, there exists a real number ε > 0 such that (a + εb, m1 ) > 0 and, (a + εb, mi ) ≥ ε(b, mi ) > 0 for i ≥ 2. We obtain that scalar multiplication (a + εb, mi ) is positive for all generators mi . Since an element m ∈ M is a linear combination of generators with natural coeﬃcients, we have (a + εb, m) > 0. For a monoid M deﬁne the set V> (M ) = {a ∈ Rn |(a, m) > 0 for every nonzero m ∈ M }. The set V> (M ) is open in Rn , and it is not empty if the monoid M ⊆ Zn is ﬁnitely generated and positive (Lemma 5.8). Let x1 , . . . , xt be pairwise distinct vectors of the space Rn . We say that a vector a ∈ Rn distinguishes the vectors x1 , . . . , xt if for every i = j we have (a, xi ) = (a, xj ). Let us prove the next lemma. Lemma 5.9. Suppose M ⊆ Zn is a ﬁnitely generated positive monoid and x1 , . . . , xt are pairwise distinct elements of M . Then there exists a vector a ∈ V> (M ) ∩ Zn n that distinguishes x1 , . . . , xt . Proof. The proof is by induction on t. For t = 2 we take an arbitrary vector a ∈ V> (M ). If (a , x1 ) = (a , x2 ), there is nothing to prove. Suppose (a , x1 ) = (a , x2 ). Since the set V> (M ) is open in Rn , there is a neighbourhood Oa of a such that Oa ⊆ V> (M ). The vector a belongs to the closure of the set X = {x|(x, x1 ) = (x, x2 )}. Since X is open, the intersection Oa ∩ X is nonempty and open in Rn . Since Oa ∩X is open, there is a vector a ∈ Oa ∩X with rational coordinates such that a distinguishes x1 and x2 . Let α be a suitable natural number such

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that the vector αa has natural coordinates. It is obvious that αa satisﬁes all conditions of the lemma. Let us proceed with the induction step. Assume that the lemma holds for all numbers t < t. Suppose x1 , . . . , xt is a set of pairwise distinct elements of M . By the inductive assumption there are vectors a1 , . . . , at ∈ V> (M ) ∩ Zn such that ai distinguishes the elements x1 , . . . , xi−1 , xi+1 , . . . , xt . We put a = a1 + A1 a2 + A1 A2 a3 , where Ai = max {(ai , xj )} + 1. 1jt

Since Ai ∈ N, we have a ∈ V> (M ) ∩ Zn . Let us prove that a distinguishes all pairs xi , xj (1 i = j t). Consider all possible cases, where integer remainders and quotients are denoted by mod and ÷ respectively. 1) If i > 1, j > 1, we have (a, xi ) mod A1 = (a1 , xi ), (a, xj ) mod A1 = (a1 , xj ). Since (a1 , xi ) = (a1 , xj ), we obtain (a, xi ) = (a, xj ). 2) If i = 1, j > 2, we have ((a, x1 ) mod A1 A2 ) ÷ A1 = (a2 , x1 ), ((a, xj ) mod A1 A2 ) ÷ A1 = (a2 , xj ). Since (a2 , x1 ) = (a2 , xj ), we obtain (a, x1 ) = (a, xj ). 3) If i = 1, j = 2, we have (a, x1 ) ÷ (A1 A2 ) = (a3 , x1 ), (a, x2 ) ÷ (A1 A2 ) = (a3 , x2 ). Since (a3 , x1 ) = (a3 , x2 ), we obtain (a, x1 ) = (a, x2 ).

5.2. Ordering of submonoids of Zn In this subsection we explain the motivation of the term “positive”. We study how for a given ﬁnitely generated positive monoid M ⊆ Zn we can deﬁne a monotone order ≤ on Zn such that this monoid will be positive with respect to ≤. An order ≤ on an abelian group G is called monotone if the following conditions hold: • ∀x∀y(x ≤ y) ∨ (y ≤ x) • ∀x, y, z(x ≤ y) ∧ (y ≤ z) → (x ≤ z) • ∀x, y, z(x ≤ y) → (x + z ≤ y + z) Note that the deﬁnition allows the existence of elements g1 , g2 ∈ G such that g1 ≤ g2 , g2 ≤ g1 , but g1 = g2 .

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Proposition 5.10 explains the choice of the term “positive monoid”. Proposition 5.10. Suppose a ﬁnitely generated positive monoid M is embedded in Zn . Then we can deﬁne a monotone order ≤ on Zn such that 0 ≤ m and m 0 for all non-trivial m ∈ M . Proof. Let M ⊆ Zn be a ﬁnitely generated positive monoid. Using Lemma 5.8 we may choose a vector a ∈ Zn such that (a, m) > 0 for all non-trivial elements of M . For all x1 , x2 ∈ Zn we put x1 ≤ x2 whenever (x1 , a) ≤ (x2 , a). It follows from the properties of scalar multiplication that the order ≤ is monotone and 0 ≤ m, m 0 whenever m ∈ M \{0}. 5.3. Proof of Theorem A Theorem A. A ﬁnitely generated monoid M is an irreducible coordinate monoid of the category CM0 (N) iﬀ M is commutative positive and has cancellation (M |= Axcomm , Axcanc , Axpos ). Proof. Let M be an irreducible coordinate monoid over N (M ∈ CM0 (N)). It follows from Theorem 1.1 that M belongs to the universal closure ucl0 (N). Therefore, the universal formulas Axcomm , Axcanc , Axpos must be true in M . Assume now that M is a ﬁnitely generated commutative positive monoid with cancellation. Let us prove that M is a coordinate monoid over N. By Theorem 1.1, it is suﬃcient to show that N discriminates M . Suppose m1 , . . . , mk are pairwise distinct elements of M . Using Proposition 5.4, we have that M is embedded into Zn . By Lemma 5.9, there is a vector a ∈ Zn ∩ V> (M ) distinguishing m1 , . . . , mk . Consider the mapping ψ(x) = (a, x). Since a ∈ Zn ∩V> (M ), we have ψ(x) ∈ N for all x ∈ M . Moreover, ψ(x + y) = ψ(x) + ψ(y) for all x, y ∈ M . Therefore, ψ : M → N is a homomorphism. Since a distinguishes m1 , . . . , mk , we obtain that ψ(mi ) = ψ(mj ) (i = j) so N discriminates M .

6. Geometric and universal equivalence It is easy to show that Theorems A and B proved for N can be extended to the subclass of commutative monoids, in particular, to the class of direct products Nk . It means that the categories CMA (N) and CMA (Nk ) are equal and, moreover, the categories of irreducible coordinate monoids over Nk and N are also equal. Therefore, N and Nk are called geometrically and universally equivalent. Below we give a deﬁnition of the geometric and universal equivalence over monoids and describe the class of monoids that are geometrically and universally equivalent to N. A-monoids M1 , M2 are called geometrically equivalent if RadM1 (S) = RadM2 (S) for every system S of equations over A. We denote the geometrical equivalence of A-monoids M1 , M2 by M1 ≈A M2 . This notion for algebraic systems was introduced by Plotkin [4]. Below we formulate the Plotkin’s problem over monoids.

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Plotkin’s problem. Find necessary and suﬃcient conditions for the geometric equivalence of A-monoids M1 , M2 . The following theorem solves this problem in the Noetherian case. Theorem 6.1 ([2], [3]). Suppose A-monoids M1 , M2 are A-equationally Noetherian. Then M1 ≈A M2 ⇐⇒ qvarA (M1 ) = qvarA (M2 ). Theorem 6.1 was proved in [2] for groups (Theorem 2) and in [3] for Lie algebras (Lemma 3.47). The proof of this theorem is universal and uses ideas from model theory. Therefore, Theorem 6.1 can be proved for monoids with the same reasoning. A-monoids M1 , M2 are called universal equivalent if M1 |= ϕ ⇔ M2 |= ϕ for every LA -formula ϕ. Denote by M1 ≈∀ M2 the universal equivalence of A-monoids M1 , M2 . Obviously, M1 ≈∀ M2 implies uclA (M1 ) = uclA (M2 ). Note that geometric equivalence does not imply universal equivalence and vice versa. The main result of this section is Theorem C, which states that a non-trivial monoid belongs to the quasivariety qvar0 (N) if and only if it is geometrically equivalent to N. We need the following two lemmas to prove Theorem C. The proof of the ﬁrst simple lemma is omitted. Lemma 6.2. Suppose M1 and M2 are A-monoids. • qvarA (M1 ) = qvarA (M2 ) iﬀ M2 ∈ qvarA (M1 ) and M1 ∈ qvarA (M2 ); • uclA (M1 ) = uclA (M2 ) iﬀ M2 ∈ uclA (M1 ) and M1 ∈ uclA (M2 ). Lemma 6.3. Suppose M ∈ qvar0 (N). Then N ∈ ucl0 (M ). Proof. Let m be a nonzero element of M . Since M is positive, the submonoid generated by m is isomorphic to N. Suppose ϕ is a universal formula such that M |= ϕ. It is clear that m N |= ϕ. Therefore we obtain N ∈ ucl0 (M ). Theorem C. A nontrivial monoid M belongs to the quasivariety qvar0 (N) iﬀ M ≈0 N. In this case all algebraic sets over M are irreducible (qvar0 (M ) = ucl0 (M )), and also M ≈∀ N. Proof. Suppose M ∈ qvar0 (N) and M = {0}. By Lemma 6.3, we have N ∈ ucl0 (M ) ⊆ qvar0 (M ). Since M ∈ qvar0 (N), using Lemma 6.2, we obtain qvar0 (N) = qvar0 (M ). It follows from Theorem 6.1 and Remark 3.9 that M ≈0 N. Suppose M ≈0 N. It easily follows from Theorem 6.1 and Lemma 6.2 that M ∈ qvar0 (N). Using Theorem B, 1.1, 3.8, we have M ∈ qvar0 (N) = ucl0 (N). By Lemma 6.3, we obtain ucl0 (M ) = ucl0 (N) = qvar0 (N) = qvar0 (M ). Therefore M ≈∀ N and all algebraic sets over M are irreducible. Corollary 6.4. A monoid M = {0} is geometrically and universally equivalent to N if and only if M is a commutative positive monoid with cancellation.

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7. Dimension theory Since all the algebraic sets of the category AS 0 (N) are irreducible (Theorem B), we give the deﬁnition of dimension an algebraic set only for irreducible algebraic sets. Let Y be an irreducible algebraic set and Y = Y0 ⊃ Y1 ⊃ · · · ⊃ Ym be a series of irreducible sets such that this series has the maximal length among all the series of this form. The length of this series is called the dimension, dim Y , of the set Y . Note that the dimension of every algebraic set is ﬁnite because N is equationally Noetherian. Suppose Y ∈ AS 0 (N) and Y ⊆ Nn . We embed Y into Rn using the natural embedding of Nn into Rn . The subspace generated by the image of Y in Rn is denoted by ρ(Y ). The main result of this section is Theorem D. Using this theorem we can easily reduce the calculation of the dimension of an algebraic set to the calculation of the dimension of the corresponding subspace of Rn . Lemma 7.1. Suppose Y1 , Y2 ∈ AS 0 (N), and ρ(Y1 ) = ρ(Y2 ), then Y1 = Y2 . Proof. Assume the converse, i.e., Y1 = Y2 . Then there is a point y ∈ Y1 \Y2 . Let e1 , . . . , em be a basis of ρ(Y1 ). Then y ∈ Nn is equal to a nontrivial linear combination of e1 , . . . , em y = α1 e1 + · · · + αm em . We can assume that α1 = 0. Hence the set of vectors y, e2 , . . . , em is also a basis of ρ(Y1 ). Since y ∈ / ρ(Y2 ), we obtain ρ(Y1 ) = ρ(Y2 ), in contradiction to the conditions of the lemma. Theorem D. For every algebraic set Y of the category AS 0 (N) we have dim Y = dim ρ(Y ). Proof. Let Y ∈ AS 0 (N) and Y = Y0 ⊃ · · · ⊃ Ym

(8)

be a chain of irreducible algebraic sets. We assume that (8) is a chain of the maximal length. We can construct the chain of subspaces ρ(Y ) = R0 ⊃ · · · ⊃ Rm ,

(9)

where Ri = ρ(Yi ). By Lemma 7.1, all the sets in (9) are pairwise distinct. Let us show that dim Ri − dim Ri+1 = 1 for i = 1, . . . , m − 1. Assume the converse, i.e., there is a number i such that dim Ri − dim Ri+1 = k > 1. Since ρ(Yi ) = Ri and ρ(Yi+1 ) = Ri+1 , we have that the sets Ri and Ri+1 are algebraic over R and are deﬁned by a systems Si , Si+1 with natural coeﬃcients. It is clear that Si ⊂ Si+1 because Yi ⊃ Yi+1 . Since k > 1, there exists a system S such that Si ⊂ S ⊂ Si+1 . The subspace R generated by solutions of S satisﬁes the conditions Ri ⊃ R ⊃ Ri+1 . The system S has natural coeﬃcients, so there is an algebraic set Y over N such that ρ(Y ) = R .

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It follows from the properties of R and Lemma 7.1 that Yi ⊃ Y ⊃ Yi+1 , but these inclusions contradict to the condition that the length of the chain (8) is maximal. We have proved that dim Ri − dim Ri+1 = 1 for i = 1, . . . , m − 1, so the chain (9) is maximal. Since the chains (8),(9) have the same length, we have Y = dim ρ(Y ). Corollary 7.2. Suppose Y = Y0 ⊃ · · · ⊃ Yn is a series of irreducible algebraic sets of AS 0 (N), such that the set inclusions are strict and the length of the series can not be increased. Then dim Y = n. Proof. Given the series Y = Y0 ⊃ · · · ⊃ Yn using the mapping ρ, we can construct the series ρ(Y ) = ρ(Y0 ) ⊃ · · · ⊃ ρ(Yn ). It is easy to prove that the length of the series constructed can not be increased. Using the properties of the space Rn we obtain that dim ρ(Y ) = n, and it follows from Theorem D that dim Y = n.

References [1] E. Daniyarova, A. Miasnikov, V. Remeslennikov, Uniﬁcation theorems in algebraic geometry, Algebra and Discrete Mathematics, 1 (2008), 137–164. [2] A. Myasnikov, V. Remeslennikov, Algebraic geometry over groups II: logical foundations, J. Algebra, 234 (2000), 225–276. [3] E. Daniyarova, Foundations of algebraic geometry over Lie algebras, Herald of Omsk University, Combinatorical methods in algebra and logic (2007), 8–39. [4] B. Plotkin, Varieties of algebras and algebraic varieties. Categories of algebraic varieties, Siberian Advances in Math., 7 (1997), 64–97. [5] The algebraic theory of semigroups, A.H. Cliﬀord and G.B. Preston. American Mathematical Society, 1961 (volume 1), 1967 (volume 2). [6] Rockafellar R.T., Convex Analysis, Princeton University Press, Princeton, N.J., 1970. Pavel V. Morar and Artem N. Shevlyakov Omsk Department of Institute of Mathematics Siberian Branch of the Russian Academy of Sciences Pevtsova st. 13 Omsk, 644099 Russia e-mail: [email protected] a [email protected]

Combinatorial and Geometric Group Theory Trends in Mathematics, 279–296 c 2010 Springer Basel AG

Some Graphs Related to Thompson’s Group F Dmytro Savchuk Abstract. The Schreier graphs of Thompson’s group F with respect to the stabilizer of 12 and generators x0 and x1 , and of its unitary representation in L2 ([0, 1]) induced by the standard action on the interval [0, 1] are explicitly described. The coamenability of the stabilizers of any ﬁnite set of dyadic rational numbers is established. The induced subgraph of the right Cayley graph of the positive monoid of F containing all the vertices of the form xn v, where n ≥ 0 and v is any word over the alphabet {x0 , x1 }, is constructed. It is proved that the latter graph is non-amenable. Mathematics Subject Classiﬁcation (2000). 20F65. Keywords. Thompson’s group, amenability, Schreier graphs, Cayley graphs.

Introduction Thompson’s group F was discovered by Richard Thompson in 1965. A lot of fascinating properties of this group were discovered later on, many of which are surveyed nicely in [CFP96]. It is a ﬁnitely presented torsion free group. One of the most intriguing open questions about this group is whether F is amenable. Originally this question was asked by Geoghegan in 1979 (see p. 549 of [GS87]) and since then dozens of papers were in some extent devoted to it. It was shown in [BS85] that F does not contain a nonabelian free subgroup and in [CFP96] that it is not elementary amenable. So the question of amenability of F is particularly important because F would be an example of a group given by a balanced presentation (two generators and two relators) of either amenable, but not elementary amenable group (the ﬁrst ﬁnitely presented example was constructed by R.Grigorchuk in [Gri98]), or non-amenable group, which does not contain a nonabelian free subgroup (the ﬁrst ﬁnitely presented example of this type was constructed by Ol’shanskii and Sapir in [OS02]). The study of the Schreier graphs of F was also partially inspired by the question of amenability of F . In particular, if any Schreier graph with respect The author was supported by NSF grants DMS-0600975 and DMS-0456185.

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to any subgroup is non-amenable the whole group F would be non-amenable. Unfortunately, all Schreier graphs we describe here are amenable which does not give any information about the amenability of F . But the knowledge about the structure of Schreier graphs provides some additional information about F itself. It happens that the described Schreier graph of the action of F on the set of dyadic rational numbers on the interval (0, 1) is closely related to the unitary representation of F in the space B(L2 ([0, 1])) of all bounded linear operators on L2 ([0, 1]). It reﬂects (modulo a ﬁnite part) the dynamics of F on the Haar wavelet basis in L2 ([0, 1]). We deﬁne the Schreier graph of a group action on a Hilbert space with respect to some basis and make this connection precise. R. Grigorchuk and S. Stepin in [GS98] reduced the question of amenability of F to the right amenability of the positive monoid P of F . Moreover, the amenability of F is equivalent to the amenability of the induced subgraph ΓP of the Cayley graph ΓF of F with respect to generating set {x0 , x1 } containing the positive monoid P . We construct the induced subgraph ΓS of ΓF containing all the vertices of the form xn v for n ≥ 0, v ∈ {x0 , x1 }∗ and prove that this graph is non-amenable. In this construction we use the realization of the elements of the positive monoid of F as binary rooted forests. The existence of this representation ˇ c was originally noted by K. Brown and developed by J. Belk in [Bel04] and Z. Suni´ ˇ in [Sun07]. It was also used by J. Donelly in [Don07] to construct an equivalent condition for amenability of F . The structure of the paper is as follows. In Section 1 the deﬁnition and the basic facts about Thompson’s group are given. Section 2 contains a description of the Schreier graph of the action of F on the set of dyadic rational numbers from the interval (0, 1). The coamenability of the stabilizers of any ﬁnite set of dyadic rational numbers is shown in Section 3. The Schreier graph of the action of F on L2 ([0, 1]) is constructed in Section 4. The last Section 5 contains a description of the subgraph ΓS of ΓP and a proof that ΓS is non-amenable. I would like to express my warm gratitude to Rostislav Grigorchuk for valuable comments and bringing my attention to Thompson’s group, and to Zoran ˇ c, who has pointed to the connection with forest diagrams, which simpliﬁed Suni´ the proofs in the last section. Also I want to thank the referee for constructive comments and suggestions that were very helpful.

1. Thompson’s group Deﬁnition 1. The Thompson’s group F is the group of all strictly increasing piecewise linear homeomorphisms from the closed unit interval [0, 1] to itself that are diﬀerentiable everywhere except at ﬁnitely many dyadic rational numbers and such that on the intervals of diﬀerentiability the derivatives are integer powers of 2. The group operation is superposition of homeomorphisms. Basic facts about this group can be found in the survey paper [CFP96]. In particular, it is proved that F is generated by two homeomorphisms x0 and x1

Some Graphs Related to Thompson’s Group F given by ⎧ ⎪ ⎨

0≤t≤ 1 ≤t≤ t− x0 (t) = 2 ⎪ ⎩ 2t − 1, 34 ≤ t ≤ 1,

1 2, 3 , 4

t 2,

1 , 4

281

⎧ t, 0 ≤ t ≤ 12 , ⎪ ⎪ ⎪ ⎨ t + 1, 1 ≤ t ≤ 3, 2 4 2 4 x1 (t) = ⎪ t − 18 , 34 ≤ t ≤ 78 , ⎪ ⎪ ⎩ 2t − 1, 78 ≤ t ≤ 1.

The graphs of x0 and x1 are displayed in Figure 1.

Ü¼

Ü½

Figure 1. Generators of F Throughout the paper we will follow the following conventions. For any two elements f , g of F and any x ∈ [0, 1] f g = gf g −1 .

(f g)(x) = g(f (x)),

(1)

With respect to the generating set {x0 , x1 } F is ﬁnitely presented. But for some applications it is more convenient to consider an inﬁnite generating set {x0 , x1 , x2 , . . .}, where n−1

xn = (x1 )x0

.

With respect to this generating set (and with respect to convention (1)) F has a nice presentation F ∼ = x0 , x1 , x2 , . . . | xk xn = xn+1 xk , 0 ≤ k < n.

(2)

2. The Schreier graph of the action of F on the set of dyadic rational numbers Let G be a group generated by a ﬁnite generating set S acting on the set M . The Schreier graph Γ(G, S, M ) of the action of G on M with respect to the generating set S is an oriented labelled graph deﬁned as follows. The set of vertices of Γ(G, S, M ) is M and there is an arrow from x ∈ M to y ∈ M labelled by s ∈ S if and only if xs = y.

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For any subgroup H of G, the group G acts on the right cosets in G/H by right multiplication. The corresponding Schreier graph Γ(G, S, G/H) is denoted as Γ(G, S, H) or just Γ(G, H) if the generating set is clear from the context. Conversely, if G acts on M transitively, then Γ(G, S, M ) is canonically isomorphic to Γ(G, S, StabG (x)) for any x ∈ M , where the vertex y ∈ M in Γ(G, S, M ) corresponds to the coset from G/ StabG (x) consisting of all elements of G that move x to y. Consider the subgroup StabF ( 12 ) of F consisting of all elements of F that ﬁx 1 1 2 . There is a natural isomorphism ψ : StabF ( 2 ) → F × F given by

t + 1 1 t ψ ( f (t) −→ 2f , 2f − 1 ∈ F × F. (3) StabF 2 2 2 This group was studied in [Bur99], where it was shown that it embeds into F quasi-isometrically. The Schreier graph Γ(F, {x0 , x1 }, StabF ( 12 )) coincides with the Schreier graph of the action of F on the orbit of 12 . Let D be the set of all dyadic rational numbers from the interval (0, 1). It is known that F acts transitively on D (which follows also from the next proposition). Therefore the latter graph coincides with the Schreier graph Γ(F, {x0 , x1 }, D). Proposition 1. The Schreier graph Γ(F, {x0 , x1 }, D) has the following structure (dashed arrows are labelled by x0 and solid arrows by x1 )

Proof. Deﬁne the following subsets of D. : 9k k is odd ∩ 1 , 3 , n≥3 An = 2n 2 4 9k : 3 7 k is odd ∩ Bn = , , n≥4 2n 4 8 1 5 5 3 , , Dn = An ∩ , , n≥4 Cn = An ∩ 2 8 8 4

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283

On the graph above, An represents the (n − 3)-rd level of the gray vertices in the binary tree; Bn is the set of the white vertices between levels n − 4 and n − 3 of the tree, which are adjacent to 2 gray vertices; Cn and Dn are the sets of the gray vertices of the (n − 3)-rd level having gray and white neighbors above respectively.

½

·½

½ ½ ½

·½

·½

¼ ½ ¼ ¼

½

Figure 2. Dynamics of x0 and x1 Now we compute the action of F on this subsets (see Figure 2). We have −1 x−1 0 (An ) = Bn+1 , x1 (Bn ) = Dn , x1 (An ) = Cn+1 , hence (x0 x1 )(An ) = Dn+1 −1 and (x0 x1 )(An ) ∪ x1 (An ) = An+1 . Therefore, we immediately obtain that the monoid generated by x1 and x−1 0 x1 is free. Indeed, any relation in this monoid would create a loop in the Schreier graph contradicting to the fact that all the sets Cn and Dn are disjoint. Hence, the grey part of the graph is a binary tree. Furthermore, for any set A ⊂ R denote αA + β = {αa + β : a ∈ A}. Then xk0 (An ) = xk0 x1 (An ) = 2−k+1 (An − 14 ) for k ≥ 1. This corresponds to the rays with the black vertices sticking out to the right from the gray ones. On the other hand since the actions of x−1 and x−1 on [ 34 , 1] coincide, for any element f of length 0 1 −1 −k k ≥ 0 from the monoid generated by x−1 (1−Bn ). 0 and x1 we have f (Bn ) = 1−2 This corresponds to the rays with white vertices. There is one more geodesic line in the graph corresponding to 12 which completes the picture. This graph gives alternative proofs of the following well-known facts. Corollary 1. (a) Thompson’s group F acts transitively on the set D of all dyadic rationals from the interval (0, 1). (b) StabF ( 12 ) acts transitively on the sets of dyadic rationals from the intervals (0, 12 ) and ( 12 , 1).

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Proof. Part (a) follows immediately from the structure of the Schreier graph F/ StabF ( 12 ). Part (b) is a consequence of part (a) and the isomorphism (3). Proposition 2. The subgroup StabF ( 12 ) is a maximal subgroup in F . Proof. Let f be any element from F \ StabF ( 12 ). Then for any g ∈ F we show that g ∈ StabF ( 12 ), f . Let g be an arbitrary element in F that does not stabilize 12 . Denote u = f ( 12 ) and v = g( 12 ). Without loss of generality we may assume 1 u < 2 . Then by transitivity from Corollary 1(b) there exists h ∈ StabF ( 12 ) such that either h(f ( 12 )) = v or h(f −1( 12 )) = v depending on whether v < 12 or v > 12 . In any case the element f˜ = f h (or f˜ = f −1 h) belongs to StabF ( 12 ), f and satisﬁes f˜( 12 ) = v. ˜ = g f˜−1 we have h( ˜ 1 ) = f˜−1 (g( 1 )) = f˜−1(v) = 1 . Thus h ˜ ∈ Now for h 2 2 2 1 1 ˜ ˜ StabF ( 2 ) and g = hf ∈ StabF ( 2 ), f . Proposition 1 also yields a bound on the length of an element. Namely, if the graph of an element f ∈ F passes through the point (a, b) for some dyadic rational numbers a and b, then the length of f with respect to the generating set {x0 , x1 } is not smaller than the combinatorial distance between a and b in the graph Γ(F, {x0 , x1 }, D). Estimates similar in spirit (also based on the properties of graph of an element, but in a diﬀerent realization of F ) were used by J.Burillo in [Bur99] to show that StabF ( 12 ) quasi-isometrically embeds into F .

3. Coamenability of stabilizers of several dyadic rationals In this section we show that for any ﬁnite subset {d1 , . . . , dn } of dyadic rationals the Schreier graphs of F with respect to StabF (d1 , . . . , dn ) is amenable. First we recall the deﬁnition of an amenable graph. Deﬁnition 2. Given an inﬁnite graph Γ = (V, E) of bounded degree the Cheeger constant h(Γ) is deﬁned as follows h(Γ) = inf S

|∂S| , |S|

where S runs over all nonempty ﬁnite subsets of V , and ∂S, the boundary of S, consists of all vertices of V \ S that have a neighbor in S. Deﬁnition 3. A graph Γ is called amenable if h(Γ) = 0. Deﬁnition 4. A subgroup H of a group G is called coamenable in G if the Schreier graph Γ(G, H) is amenable. Note, that coamenability of a subgroup does not depend on the generating set of G. This follows easily from Gromov’s doubling condition (see Theorem 5 in Section 5).

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Proposition 3. Let {d1 , . . . , dn } ⊂ D be any ﬁnite subset of dyadic rationals. Then the subgroup StabF (d1 , . . . , dn ) of F consisting of all elements stabilizing all the di ’s is coamenable in F . Proof. First, we describe the structure of the Schreier graph Γ(F, {x0 , x1 }, StabF (d1 , . . . , dn )), d1 < d2 < · · · < dn . Analogously to the singleton case there is a one-to-one correspondence between cosets from F/ StabF (d1 , . . . , dn ) and all strictly increasing n-tuples of dyadic rationals. This follows from the fact that F acts transitively on the latter set (see [CFP96]). There is an edge labelled by s ∈ {x0 , x1 } from the coset (d1 , . . . , dn ) to the coset (d1 , . . . , dn ) if and only if s(di ) = di for every i. Geometrically one can interpret this in the following way. Consider a disjoint union of n copies of Γ(F, {x0 , x1 }, StabF ( 12 )) (a layer for each di ). Then the coset (d1 , . . . , dn ) of F/ StabF (d1 , . . . , dn ) can be represented by the path joining di vertex on the ith layer with di+1 vertex on the (i + 1)th layer (see Figure 3). The action of the generators on the set of such paths is induced by the independent actions of the generators on the layers.

Figure 3. Cosets in F/ StabF (d1 , d2 , d3 )

Now deﬁne Ei = and

1

2

, i+n

1

2

,..., i+n−1

1 2i+1

∈ F/ StabF (d1 , . . . , dn )

Sm = Ei 1 ≤ i ≤ m .

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Since x1 (Ei ) = Ei and x0 (Ei ) = Ei+1 we have that the boundary ∂Sm = {E0 , Em+1 } and lim

m→∞

|∂Sm | 2 = lim = 0. m→∞ m |Sm |

Thus h(Γ(F, {x0 , x1 }, StabF (d1 , . . . , dn ))) = 0 and StabF (d1 , . . . , dn ) is coamenable in F . The amenability of the action of F on the set of dyadic rational numbers and on the set of the ordered tuples of dyadic rational numbers was also noted independently by N. Monod and Y. Glasner (private communication).

4. The Schreier graph of the action of F on L2 ([0, 1]) There is a natural unitary representation of Thompson’s group F in the space B(L2 ([0, 1])) of all bounded linear operators on L2 ([0, 1]). For g ∈ F and f ∈ L2 ([0, 1]) deﬁne ; dg(x) f (g −1 x), (πg f )(x) = dx where dg(x) dx denotes the Radon-Nikodym derivative of the measure on [0, 1] induced by g with respect to Lebesgue measure. In this case it coincides with a regular derivative of a real-valued function g. For the detailed explanation we refer the reader to [BdlHV08, Appendix A.6]. For our purposes it is convenient to consider this action with respect to the (i) orthonormal Haar wavelet basis B = {h(0) , hj , i ≥ 0, j = 1 . . . 2i } in L2 ([0, 1]), where h(0) (x) ≡ 1 and −1, x < 12 , (0) h1 (x) = 1, x ≥ 12 , ⎧ i j−1 j−1 1 ⎪ ⎨ −2 2 , 2i ≤ x < 2i + 2i+1 , i (i) j 1 hj (x) = 2 2 , j−1 2i + 2i+1 ≤ x ≤ 2i , ⎪ ⎩ j 0, x ∈ / [ j−1 2i , 2i ]. This basis has ﬁrst appeared in 1910 in the paper of Haar [Haa10] and plays an important role in the wavelet theory (see, for example, [Dau92, WS01]). The convenience of using this basis for us comes from the following fact. (i) Each of the generators x0 and x1 acts on each of the basis functions hj for i ≥ 3 (i)

linearly on the support of hj , so that the image also belongs to B. More precisely,

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straightforward computations yield (i)

(i+1)

(i)

(i)

(i)

(i−1)

(i)

(i)

i ≥ 1, 1 ≤ j ≤ 2i−1 ,

(i)

(i+1)

i ≥ 2, 2i−1 + 1 ≤ j ≤ 2i−1 + 2i−2 ,

(i)

(i)

(i)

(i−1)

πx0 hj = hj

,

i ≥ 1, 1 ≤ j ≤ 2i−1 ,

πx0 hj = hj−2i−2 , i ≥ 2, 2i−1 + 1 ≤ j ≤ 2i−1 + 2i−2 , πx0 hj = hj−2i−1 , i ≥ 2, 2i−1 + 2i−2 + 1 ≤ j ≤ 2i , πx 1 h j = h j , πx1 hj = hj+2i−1 ,

πx1 hj = hj−2i−3 , i ≥ 3, 2i−1 + 2i−2 + 1 ≤ j ≤ 2i−1 + 2i−2 + 2i−3 , πx1 hj = hj−2i−1 , i ≥ 3, 2i−1 + 2i−2 + 2i−3 + 1 ≤ j ≤ 2i .

(4)

There is a one-to-one correspondence ψ between B \ {h(0) } and the set of (i) 1 all dyadic rationals from the interval (0, 1) given by ψ(hj ) = j−1 + 2i+1 , that 2i is, each basis function corresponds to the point of its biggest jump (where the function changes the sign). Below we will use the following simple observation, which can also be used to derive equalities (4). If a function h(x) ∈ L2 ([0, 1]) changes its sign at the point x0 then for any g ∈ F the function (πg h)(x) changes its sign at the point g(x0 ). (i) This enables us to ﬁnd the image of hj , i ≥ 3 under action of πxk , k = 0, 1 in the following easy way: (i) (i) πxk hj = ψ −1 xk (ψ(hj )) In other words the following diagram is commutative for k = 0, 1 (i)

hj ⏐ ⏐ ψ$ j−1 2i

+

1 2i+1

πx

k −−−− →

−−−−→ xk

(i )

hj ⏐ ⏐ ψ$ j −1 2i

+

1 2i +1

Now we deﬁne the Schreier graph of the action of a group on a Hilbert space. Let H be a Hilbert space with an orthonormal basis {hi , i ≥ 1}. Suppose there is a representation π of a group G = S in the space of all bounded linear operators B(H). We denote the image of g ∈ G under π as πg . Deﬁnition 5. The Schreier graph Γ of the action of a group G on a Hilbert space H with respect to the basis {hi , i ≥ 1} of H and generating set S ⊂ G is an oriented labelled graph deﬁned as follows. The set of vertices of Γ is the basis {hi , i ≥ 1} and there is an arrow from hi to hj with label s ∈ S if and only if πs (hi ), hj = 0 (in other words the coeﬃcient of πs (hi ) at hj in the basis {hi , i ≥ 1} is nonzero). The argument above shows that the Schreier graph of the Thompson’s group action on L2 ([0, 1]) with respect to the Haar basis and generating set {x0 , x1 } coincides modulo a ﬁnite part with the Schreier graph Γ(F, {x0 , x1 }, D). In order

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to complete the picture we have to ﬁnd the images under the action of πx0 and (i) πx1 of those hj which are not listed in (4). Again straightforward computations give the following equalities. / / √ 0 √ 0 1 1 (0) 1 2 2 (1) (0) (0) + h − h1 + − + h1 , πx0 h = 4 2 4 2 4 / / √ 0 √ 0 1 (0) 1 1 2 2 (0) (0) (1) πx0 h1 = h + − + h1 + + h1 , 4 4 2 2 4 / / √ 0 √ 0 1 1 1 (1) 2 2 (1) (0) (0) − h + + h1 − h1 , πx0 h2 = 2 4 2 4 2 / / / √ √ 0 √ 0 √ 0 5 3 1 2 2 2 (1) 2 (0) (2) (0) (0) πx1 h = + h + − + h1 − h + − h3 , 8 4 8 4 8 2 4 4 / / / √ 0 √ 0 √ 0 √ 3 5 1 2 2 2 2 (0) (0) (1) (2) πx1 h1 = − + h(0) + + h1 − h + − h3 , 8 4 8 4 8 2 4 4 / / √ √ √ 0 √ 0 1 1 2 (0) 2 (0) 2 2 (1) (1) (2) h + h + − + h2 + + h3 , πx1 h1 = 8 8 1 4 2 2 4 / / / √ 0 √ 0 √ 0 1 1 1 1 (2) 2 2 2 (2) (0) (1) πx 1 h 4 = − + h(0) + − + h1 + + h 2 − h3 . 4 4 4 4 2 4 2 These computations together with Proposition 1 prove the following proposition. Proposition 4. The Schreier graph of Thompson’s group action on L2 ([0, 1]) with respect to the Haar basis and the generating set {x0 , x1 } has the following structure (dashed arrows are labelled by x0 and solid arrows by x1 )

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289

5. Parts of the Cayley graph of F Recall, that the positive monoid P of F is the monoid generated by all generators xi , i ≥ 0. As a monoid it has a presentation P ∼ = x0 , x1 , x2 , . . . | xk xn = xn+1 xk , 0 ≤ k < n, which coincides with the inﬁnite presentation (2) of F . The group F itself can be deﬁned as a group of left fractions of P (i.e., F = P −1 · P ). It was shown in [GS98] (see also [Gri90]) that the amenability of F is equivalent to the right amenability (with respect to our convention (1)) of P . Moreover, let ΓF be the Cayley graph of F with respect to the generating set {x0 , x1 } and ΓP be the induced subgraph of ΓF containing positive monoid P . The following proposition is of a folklore type. Proposition 5. Amenability of F is equivalent to amenability of the graph ΓP . Proof. Any ﬁnite set T in F can be shifted to the positive monoid P , i.e., there is some g ∈ F such that T g ⊂ P . The boundary ∂P (T g) of this shifted set in ΓP is not bigger than the boundary of T in ΓF . Hence, Cheeger constant of ΓP is not bigger than the one of ΓF . Thus, non-amenability of ΓP implies non-amenability of F . Suppose that ΓP is amenable. Then for any ε > 0 there exists a subset T of P , such that its boundary ∂P T in ΓP satisﬁes ε |∂P T | < . |T | 4

(5)

Now we can bound the size of the boundary ∂F T of T in ΓF . We use simple observations that for ﬁnite sets A and B of the same cardinality |A\B| = |B \A| = −1 −1 1 2 |AΔB| and that |T xi ΔT | = |(T xi ΔT )xi | = |T xi ΔT |. We have −1 ∂F T = (T x0 \ T ) ∪ (T x1 \ T ) ∪ (T x−1 0 \ T ) ∪ (T x1 \ T ).

Therefore, −1 |∂F T | ≤ |T x0 \ T | + |T x1 \ T | + |T x−1 0 \ T | + |T x1 \ T | 1 −1 = (|T x0 ΔT | + |T x1 ΔT | + |T x−1 0 ΔT | + |T x1 ΔT |) 2 = |T x0 ΔT | + |T x1 ΔT | = 2|T x0 \ T | + 2|T x1 \ T | ≤ 4|∂P T | < ε|T |

since T xi \ T ⊂ ∂P T for i = 1, 2 and by (5). This shows that ΓF is also amenable in this case. In this section we explicitly construct the induced subgraph ΓS of ΓF containing the set of vertices S = {xn u n ≥ 0, u is a word over {x0 , x1 }}. (6) We also prove that this graph is non-amenable.

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D. Savchuk

Since S is included in the positive monoid of F and contains elements from the inﬁnite generating set {x0 , x1 , x2 , . . .}, it is natural to use the language of forest ˇ diagrams developed in [Bel04, Sun07] (though the existence of this representation was originally noted by K.Brown [Bro87]). First we recall the deﬁnition and basic facts about this representation of the elements of F . A binary forest is an ordered sequence of ﬁnite rooted binary trees (some of which may be trivial). The forest is called bounded if it contains only ﬁnitely many nontrivial trees. There is a one-to-one correspondence between the elements of the positive monoid of F and bounded rooted binary forests. More generally, there is a one-toone correspondence between elements of F and, so-called, reduced forest diagrams, but for our purposes (and for simplicity) it is enough to consider only the elements of the positive monoid.

There is a natural way to enumerate the leaves of the trees in the forest from left to right. First we enumerate the leaves of the ﬁrst tree from left to right, then the leaves of the second tree, etc. Also there is a natural left-to-right order on the set of the roots of the trees in the forest. The product fg of two rooted binary forests f and g is obtained by stacking the forest g on the top of f in such a way, that the ith leaf of g is attached to the ith root of f. For example, if g and f have the following diagrams

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291

then their product fg is the following rooted binary forest

With this operation the set of all rooted binary forests is isomorphic (see ˇ [Bel04, Sun07]) to the positive monoid of Thompson’s group F , where xn corresponds to the forest in which all the trees except the (n + 1)st one (which has number n) are trivial and the (n + 1)st tree represents a single caret. Below is the picture of the forest corresponding to x3 .

The multiplication rule for the forests implies the following algorithm for construction of the rooted forest corresponding to the element xi1 xi2 xi3 · · · xin of the positive monoid of F . Start from the trivial forest (where all the trees are singletons) and consequently add the carets at the positions i1 , i2 ,. . . , in (counting from 0 the roots of the trees in the forest in previous iteration). For our main result in this section we need two lemmas. Lemma 1. Let u be a word from the positive monoid of the form u = xn v, where n ≥ 2 and v is a word over the alphabet {x0 , x1 } of length at most n − 2. Then this word is not equal in F to any other word of the form xm w, where w is a word over {x0 , x1 }. Proof. The forest diagram corresponding to u has a caret c connecting the nth and (n + 1)st leaves corresponding to xn and possibly some nontrivial trees to the left of c.

¼

½

¾

¾

½

·½

·¾

Figure 4. Forest corresponding to xn v Indeed, after attaching the caret corresponding to xn all the other carets are attached at positions either 0 or 1. Each of these carets decreases the number of

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D. Savchuk

trees to the left of caret c by 1. Since originally there were n trees to the left from c and the length of v is at most n − 2, there must be at least 2 trees to the left of c in the forest representing u. Suppose there is another word of the form xm w in the positive monoid of F whose corresponding rooted forest coincides with the forest of u. Since there are at least 2 trees to the left of caret c one cannot obtain this caret by applying x0 or x1 . Therefore it was constructed at the ﬁrst step with application of xm . Thus xm = xn because this caret connects the nth and (n + 1)st leaves, which, in turn, implies that v = w in F . But both v and w are the elements of a free submonoid generated by x0 and x1 , yielding that xn v = xm w as words. Lemma 2. Let u be a word from the positive monoid of the form u = xn vx1 v , where n ≥ 2, v is a word over the alphabet X = {x0 , x1 } of length n − 2 and v is a word over the alphabet X = {x0 , x1 } of arbitrary length. Then this word is not equal in F to any other word of the form xm w, where w is a word over {x0 , x1 }. Proof. The rooted forest corresponding to xn v is constructed in Lemma 1 and shown in Figure 4. Note, that there are exactly 2 trees (one of which is shown trivial in Figure 4 but generally both can be nontrivial) to the left of caret c. At the next step we apply generator x1 , which attaches the new caret d that connects the root of the second of these trees to the root of caret c. The resulting forest is shown in Figure 5.

¼

½

¾

¾

½

·½

·¾

Figure 5. Forest corresponding to xn vx1 Next, applying v adds some extra carets on top of the picture. The ﬁnal rooted forest is shown in Figure 6. Analogously to Lemma 1 we obtain that if the rooted forest of xm w coincides with the one of u, the caret c could appear only from the initial application of xm (since it must be placed before caret d is placed). Hence xn = xm and v = w as words, because the submonoid generated by x0 and x1 is free. Let ΓS be the induced subgraph of the Cayley graph ΓF of F that contains all the vertices of from the set S (recall the deﬁnition of S in (6)). As a direct corollary of Lemma 1 and Lemma 2, we can describe explicitly the structure of ΓS (see Figure 7, where solid edges are labelled by x1 and dashed by x0 ).

Some Graphs Related to Thompson’s Group F

¼

½

¾

¾

½

·½

·¾

Figure 6. Forest corresponding to xn vx1 v

Figure 7. Induced subgraph ΓS of the Cayley graph of F

293

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D. Savchuk

Proposition 6. The structure of ΓS is as follows (a) ΓS contains the inﬁnite binary tree T corresponding to the free submonoid generated by x0 and x1 ; (b) for each n ≥ 2 there is a binary tree Tn in ΓS consisting of n − 2 levels which grows from the vertex xn and does not intersect anything else; (c) Each vertex xn v of the boundary of Tn (i.e., v has length n − 2) has two neighbors xn vx1 and xn vx0 outside Tn . The ﬁrst one is the root of an inﬁnite binary tree which does not intersect anything else. The second one coincides with the vertex vx0 x1 of the binary tree T . Proposition 7. The graph ΓS is non-amenable. In order to prove this proposition we will use equivalent to the amenability doubling condition (or Gromov doubling condition) [dlAGCS99]. Theorem A (Gromov’s Doubling Condition). Let X be a connected graph of bounded degree. Then X is non-amenable if and only if there is some k ≥ 1 such that for any ﬁnite nonempty subset S ⊂ V (X) we have |Nk (S)| ≥ 2|S|, where Nk (S) is the set of all vertices v of X such that dX (v, S) ≤ k. Proof of Proposition 7. In order to use Theorem 5 it is enough to construct two injective maps f, g : V (X) → V (X) with distinct images, that do not move vertices farther than by distance k. For any vertex xn v in S put f (xn v) = xn vx1 x0 , g(xn v) = xn vx1 x1 . For any vertex xn v of S we have d(xn v, f (xn v)) = 2 and d(xn v, g(xn v)) = 2, so the last condition of Theorem 5 is satisﬁed. The relation f (xn v) = f (xm w) implies xn vx1 x0 = xm wx1 x0 and xn v = xm w. Hence f is an injection. The same is true for g. Now suppose f (xn v) = g(xm w) or, equivalently, xn vx1 x0 = xm wx1 x1 .

(7)

The words xn vx1 and xm wx1 represent diﬀerent vertices in ΓS since otherwise we would get x0 = x1 . According to Proposition 6 the equality (7) is possible only in case when xn vx1 is a vertex of the boundary of Tn and xm wx1 is a vertex of T . But by Proposition 6(c) in this case the vertex xn vx1 x0 coincides with the vertex vx1 x0 x1 of T which cannot coincide with xm wx1 x1 . Indeed, otherwise we get vx1 x0 = xm wx1 , which means that xm wx1 is a vertex of T . The only way a vertex of the form ux1 belongs to T is if u is in T . Therefore the last equality may not occur because T is a tree. Thus by Theorem 5 the graph ΓS is non-amenable.

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References [BdlHV08]

Bachir Bekka, Pierre de la Harpe, and Alain Valette. Kazhdan’s property (T ), volume 11 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2008.

[Bel04]

James M. Belk. Thompson’s group F . PhD thesis, Cornell University, 2004.

[Bro87]

Kenneth S. Brown. Finiteness properties of groups. In Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985), volume 44, pages 45–75, 1987.

[BS85]

Matthew G. Brin and Craig C. Squier. Groups of piecewise linear homeomorphisms of the real line. Invent. Math., 79(3):485–498, 1985.

[Bur99]

Jos´e Burillo. Quasi-isometrically embedded subgroups of Thompson’s group F . J. Algebra, 212(1):65–78, 1999.

[CFP96]

J.W. Cannon, W.J. Floyd, and W.R. Parry. Introductory notes on Richard Thompson’s groups. Enseign. Math. (2), 42(3-4):215–256, 1996.

[Dau92]

Ingrid Daubechies. Ten lectures on wavelets, volume 61 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.

[dlAGCS99] P. de lya Arp, R.I. Grigorchuk, and T. Chekerini-Sil bersta˘ın. Amenability and paradoxical decompositions for pseudogroups and discrete metric spaces. Tr. Mat. Inst. Steklova, 224 (Algebra. Topol. Diﬀer. Uravn. i ikh Prilozh.):68–111, 1999. [Don07]

John Donnelly. Ruinous subsets of Richard Thompson’s group F . J. Pure Appl. Algebra, 208(2):733–737, 2007.

[Gri90]

R.I. Grigorchuk. Growth and amenability of a semigroup and its group of quotients. In Proceedings of the International Symposium on the Semigroup Theory and its Related Fields (Kyoto, 1990), pages 103–108, Matsue, 1990. Shimane Univ.

[Gri98]

R.I. Grigorchuk. An example of a ﬁnitely presented amenable group that does not belong to the class EG. Mat. Sb., 189(1):79–100, 1998.

[GS87]

S.M. Gersten and John R. Stallings, editors. Combinatorial group theory and topology, volume 111 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1987. Papers from the conference held in Alta, Utah, July 15–18, 1984.

[GS98]

R.I. Grigorchuk and A.M. Stepin. On the amenability of cancellation semigroups. Vestnik Moskov. Univ. Ser. I Mat. Mekh., (3):12–16, 73, 1998.

[Haa10]

Alfred Haar. Zur Theorie der orthogonalen Funktionensysteme. Math. Ann., 69(3):331–371, 1910.

[OS02]

Alexander Yu. Ol shanskii and Mark V. Sapir. Non-amenable ﬁnitely pre´ sented torsion-by-cyclic groups. Publ. Math. Inst. Hautes Etudes Sci., (96):43–169 (2003), 2002. ˇ c. Tamari lattices, forests and Thompson monoids. European J. Zoran Suni´ Combin., 28(4):1216–1238, 2007.

ˇ [Sun07]

296 [WS01]

D. Savchuk Gilbert G. Walter and Xiaoping Shen. Wavelets and other orthogonal systems. Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton, FL, 2001.

Dmytro Savchuk Department of Mathematics Texas A&M University College Station, TX 77843-3368, USA e-mail: [email protected]

Combinatorial and Geometric Group Theory Trends in Mathematics, 297–305 c 2010 Springer Basel AG

Generating Tuples of Virtually Free Groups Richard Weidmann Abstract. We give a complete description of all generating tuples of a virtually free group, i.e., we give a parametrization of Epi(Fn , Γ) where n ∈ N and Γ is a virtually free group. Mathematics Subject Classiﬁcation (2000). 20F05, 20F67. Keywords. Nielsen equivalence, virtually free groups.

1. Introduction For a free group of rank k ≤ n it is the fundamental result of J. Nielsen [N] that says that for any two epimorphisms φ1 , φ2 : Fn → Fk there exists an automorphism α ∈ Aut Fn such that φ1 = φ2 ◦ α. This is equivalent to saying that any two generating n-tuples of Fk are Nielsen equivalent. The analogous statement is true for free Abelian groups and some ﬁnite groups. By the proof of Grushko’s theorem the property of having unique Nielsen classes of generating tuples is further closed under free products. Uniqueness of the Nielsen class of minimal generating tuples of surface groups has been established by Zieschang [Z]. While Nielsen classes are often not unique there are still a number of situations where the ﬁniteness of the number of Nielsen classes can be established. We restrict our discussion to negatively curved groups. Delzant [Del] proves that torsion-free hyperbolic groups have only ﬁnitely many Nielsen classes of generating pairs. This has been generalized to tuples of arbitrary size provided that the group is locally quasiconvex [KW1] or Kleinian [KW2]. It has further recently been shown that torsion-free (non-hyperbolic) Kleinian groups can have inﬁnitely many Nielsen classes of generating pairs [HW]. It turns out that the torsion-freeness assumption cannot be dropped in the above statements. Indeed there are 2-generated virtually free groups that have inﬁnitely many Nielsen equivalence classes of generating pairs, see Section 6. The main purpose of this article is to show that there is nevertheless a ﬁnite description of the set of all generating tuples of a given virtually free group. This gives

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an equivalence relation that is coarser but still related to Nielsen equivalence. The picture formally resembles the Makanin-Razborov diagrams describing homomorphisms from a ﬁnitely presented group to a free group, see [R], [KM], [S]. Theorem 1.1. For any ﬁnitely generated virtually free group Γ and n ≥ rank Γ there exists a ﬁnite directed rooted tree T with root v0 satisfying 1. Any vertex v ∈ V T is labelled by a virtually free group Gv and Gv0 = Fn 2. Any valence 1 vertex diﬀerent from v0 is labeled by Γ 3. Any edge e ∈ ET is labeled by an epimorphism πe : Gα(e) → Gω(e) where α(e) is the initial and ω(e) is the terminal vertex of the edge e. such that for any epimorphism φ : Fn → Γ there exists a reduced directed path e1 , . . . , ek from v0 to some valence 1 vertex ω(ek ) such that φ = αk ◦ πek ◦ αk−1 ◦ · · · ◦ α1 ◦ πe1 ◦ α0 where α0 ∈ Aut Fn and αi ∈ Aut Gω(ei ) for 1 ≤ i ≤ k. It follows from the proof, see also Section 5, that the description is eﬀective, i.e., that the tree T and its labels can be eﬀectively computed. The paper is organized as follows. In Section 2 we brieﬂy discuss virtually free groups. In Section 3 we discuss the simple concept of the weakly reduced core of a graph of groups before giving the proof of Theorem 1.1 in Section 4. We then brieﬂy discuss a solution to the rank problem for virtually free groups in Chapter 5. We conclude by showing that virtually free groups can have inﬁnitely many T -systems which is equivalent to saying that there are examples of virtually free groups for which the tree T in Theorem 1.1 cannot be chosen such that any vertex is in distance one from the root. I would like to thank Bernhard M¨ uhlherr for pointing out that SL2 (p) provides examples for Lemma 6.3.

2. Virtually free groups A group is called virtually free if it contains a free subgroup of ﬁnite index. There are many characterizations of virtually free groups; the following is due to Karrass, Pietrowski and Solitar [KPS]. Their proof relies heavily on Stallings’ theorem on ends [St]. Theorem 2.1. A ﬁnitely generated group is virtually free iﬀ it is the fundamental group of a ﬁnite graph of ﬁnite groups. Note that free groups are residually ﬁnite and that a group G with ﬁnite index subgroup H is residually ﬁnite iﬀ H is residually ﬁnite. As residually ﬁnite groups are further hopﬁan by Malcev’s theorem [M] we have the following: Lemma 2.2. Finitely generated virtually free groups are residually ﬁnite and therefore hopﬁan.

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3. The reduced core of a graph of groups For the remainder of this article all graphs of groups are assumed to have ﬁnite underlying graph. We will follow the notational conventions from [KMW]. Recall that a graph of groups A is called minimal if the corresponding BassSerre tree contains no proper π1 (A)-invariant subtree; this is equivalent to saying that A contains no valence 1 vertex v such that the boundary monomorphism αe : Ae → Aα(e) = Av is surjective where e is the unique edge with α(e) = v. We further call a graph of groups with a basepoint (A, v0 ) minimal if A contains no valence 1 vertex v = v0 such that the boundary monomorphism αe : Ae → Aα(e) = Av is surjective where e is the unique edge with α(e) = v. The pair (A, v0 ) is further called weakly reduced if it contains no valence 2 vertex v = v0 such that such that both αe : Ae → Aα(e) = Av and αf : Af → Aα(f ) = Av are surjective where e and f are the two edges with α(e) = α(f ) = v. Note a graph of groups that is not weakly reduced can be obtained from another graph of groups with fewer edges by performing edge subdivisions. We can now assign to any pair (A, v0 ) its weakly reduced core rcore(A, v0 ) in the obvious way by ﬁrst dropping all subdivisions at vertices diﬀerent from v0 to make it weakly reduced and then dropping all inessential valence 1 vertices diﬀerent from v0 . For any pair (A, v0 ) we denote the obvious isomorphism π1 (rcore(A, v0 ), v0 ) → π1 (A, v0 ) by θA,v0 . The following result is essentially due to Linnell [L], it bounds the complexity of the weakly reduced core provided there is a uniform bound on the edge groups. The proof that gives the below bound is due to Dunwoody [D]; see [W] for a statement where the following can be immediately extracted from. Theorem 3.1 (Linnell). Let (A, v0 ) be a graph of groups with base point such that all edge groups are of order at most C and that π1 (A) is r-generated. Then rcore(A, v0 ) has at most C · r · 3 edges.

4. The proof of the theorem Throughout this section we assume that Γ is a ﬁxed ﬁnitely generated virtually free group and that n ≥ rank Γ. We choose a reduced minimal ﬁnite graph of ﬁnite groups A and a base vertex v0 ∈ V A such that Γ = π1 (A, v0 ). Recall that an A-graph (B, u0 ) encodes a morphism from some graph of groups B to A such that u0 is mapped to v0 and the morphism is injective on vertex groups; see [KMW] for details. Fn is assumed to be the free group in x1 , . . . , xn . Lemma 4.1. There exists a ﬁnite set of ﬁnite graphs of ﬁnite groups with base vertices (A1 , w1 ), . . . , (Ak , wk ) such that the following hold: If (B, u0 ) is an A-graph such that π1 (B, u0 ) is n-generated then rcore(B, u0 ) is isomorphic to some (Ai , wi ).

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Proof. This follows immediately from Theorem 3.1 and the uniform bound on the orders of all vertex groups of A (and therefore B). Note that one pair in this list, say (A1 , w1 ), must be the graph with a single vertex w1 and n loop edges e1 , . . . , en (and all vertex and edge groups trivial). Here we can choose an isomorphism β0 : Fn → π1 (A1 , w1 ) mapping the basis element xi to the homotopy class represented by ei . There further is a pair, say (Ak , wk ), that is isomorphic to (A, v0 ). We ﬁx an isomorphism between (Ak , wk ) and (A, v0 ) with induced isomorphism βt : π1 (Ak , wk ) → π1 (A, v0 ) = Γ. Note that both β0 and βt are only unique up to an automorphism induced by an automorphism of (A1 , w1 ) and (Ak , wk ), respectively. The proof of Theorem 1.1 now follows immediately from the following two lemmata. The ﬁrst lemma shows that we can ﬁnd a tree T of ﬁnite diameter that has the properties demanded in Theorem 1.1, the second one implies that we can choose the tree to be locally ﬁnite. Together they clearly imply the assertion of Theorem 1.1. Lemma 4.2. For any epimorphism φ : Fn → Γ there exists a sequence (A1 , w1 ) = (Ai1 , wi1 ), . . . , (Aim , wim ) = (Ak , wk ) of pairwise distinct pairs (from Lemma 4.1) and epimorphisms πj : π1 (Aij , wij ) → π1 (Aij+1 , wij+1 ) for 1 ≤ j ≤ m − 1 such that φ = βt ◦ αm ◦ πm−1 ◦ · · · ◦ α2 ◦ π1 ◦ α1 ◦ β0 where αj ∈ Aut π1 (Aij , wij ) for 1 ≤ j ≤ m.

Proof. Let γi = φ(xi ) for 1 ≤ i ≤ n, i.e., the epimorphism φ : Fn → Γ is given by xi → γi . We represent the γi by not necessarily reduced non-trivial A-paths pi based at vk . Thus [pi ] = γi for 1 ≤ i ≤ n, here [pi ] is the homotopy class of pi . We deﬁne (B0 , u0 ) to be the S-wedge corresponding to S = (p1 , . . . , pn ). Thus B0 is the A-graph that is a wedge of n circles (possibly subdivided into many edges) with trivial edge and vertex groups such that the loops are labeled by the pi . Note that rcore(B0 , u0 ) is isomorphic to (A1 , w1 ). Clearly π1 (B0 , u0 ) is isomorphic to Fn and we can choose an isomorphism χ : Fn → π1 (B0 , u0 ) such that φ = ν0 ◦ χ where ν0 : π1 (B, u0 ) → π1 (A, v0 ) = Γ is the homomorphism induced by the Agraph (B0 , u0 ). Let now (B0 , u0 ), . . . , (Bl , ul ) be a sequence of A-graphs such that (Bi+1 , ui+1 ) is obtained from (Bi , ui ) by a Stallings fold and that (Bl , ul ) = (A, v0 ). Such a sequence exists as all edge groups of A are ﬁnite; this even implies that any folding

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sequence terminates and has this property. Note, that all π1 (Bi , ui ) are ﬁnitely generated virtually free groups as they are fundamental groups of ﬁnite graphs of ﬁnite groups. For i = 1, . . . , l the fold induces an epimorphism ψi : π1 (Bi−1 , ui−1 ) → π1 (Bi , ui ) whose product recovers ν0 , i.e., we have ν0 = ψl ◦ · · · ◦ ψ2 ◦ ψ1 . For each i choose ji such that rcore(Bi , ui ) is isomorphic to (Aji , wji ). Further choose an isomorphism γi : π1 (rcore(Bi , ui ), ui ) → π1 (Aji , wji ). It follows that for each i we have an isomorphism ηi := γi ◦ θB−1 : π1 (Bi , ui ) → π1 (Aji , wji ). i ,ui Thus we have epimorphisms −1 ψ¯i := ηi ◦ ψi ◦ ηi−1 : π1 (Aji−1 , wji−1 ) → π1 (Aji , wji )

and we get φ = ηl−1 ◦ ψ¯l ◦ · · · ◦ ψ¯2 ◦ ψ¯1 ◦ η1 ◦ χ. Now as ηl−1 = βt ◦ α for some α ∈ Aut(π1 (Ak , wk )) and η1 ◦ χ = α ◦ β0 for some α ∈ Aut(π1 (A1 , w1 )) we have φ = βt ◦ α ◦ ψ¯l ◦ · · · ◦ ψ¯2 ◦ ψ¯1 ◦ α ◦ β0 . If now ji = jk for some i < k then ψ¯k ◦ · · · ◦ ψ¯i+1 : π1 (Aij , wij ) → π1 (Aik , wik ) is a surjective endomorphism and therefore an automorphism because of the Hopf property (Theorem 2.1). The assertion of the lemma now follows by replacing maximal subsequence of type ψ¯k ◦ · · · ◦ ψ¯i+1 that connect A-graphs with isomorphic weakly reduced cores by the corresponding automorphisms and relabel the remaining ψ¯i as πj for appropriate i and j. Lemma 4.3. For each i and j there exist up to precomposition with some α1 ∈ Aut π1 (Ai , wi ) and postcomposition with some α2 ∈ Aut π1 (Aj , wj ) only ﬁnitely many epimorphisms π1 (Ai , wi ) → π1 (Aj , wj ) that can occur as some ψ¯k in the proof of Lemma 4.1. Proof. Note ﬁrst that a fold applied to one of the Bi induces a fold on the Aji that is preceded by at most two subdivisions to make the fold simplicial on the level of the Ai , this is discussed in much detail in [D]. The claim of the lemma is now an immediate consequence of the ﬁniteness of the underlying graphs and the uniform bound on the orders of all edge groups and vertex groups involved.

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5. The rank problem Using the proof of Theorem 1.1 it is fairly simple to extract a solution to the rank problem for virtually free groups, i.e., to describe an algorithm that computes the minimal number of generators of a given virtually free group Γ. The discussion below in fact implies that the tree described in Theorem 1.1 can be algorithmically constructed. Indeed ﬁrst write Γ as the fundamental group of a ﬁnite graph of ﬁnite groups, such a decomposition can be found by enumerating presentations of Γ by enumerating Tietz transformations until there is a presentation from which one can read oﬀ a presentation of Γ as a fundamental group of a ﬁnite graph of ﬁnite groups A. It is then easy to transform A into a minimal and reduced graph of groups. Note that there are only ﬁnitely many ﬁnite graphs of minimal weakly reduced graphs of groups of rank k provided there is a uniform bound on the orders of vertex groups. These can all be constructed by enumerating folding sequences starting with the wedge of k circles as in the proof of Theorem 1.1. One can then check for each k whether A is in the list.

6. Nielsen equivalence and T-systems In this section we show that the tree T of Theorem 1.1 cannot always be chosen such that any vertex is in distance 1 from the base vertex v0 . Recall that two generating n-tuples of a group Γ are called Nielsen-equivalent if they are the images of two bases of Fn under a ﬁxed epimorphism φ. As any two bases of Fn are related by an automorphism of Fn it follows that Γ has only ﬁnitely many Nielsen-equivalence classes of generating n-tuples if there exist ﬁnitely many epimorphisms φ1 , . . . , φm : Fn → Γ such that for any epimorphism φ : Fn → Γ there exists some α ∈ Aut Fn and some i such that φ = φi ◦ α. A coarser equivalence relation is that of Tn -systems. Two generating n-tuples T1 and T2 of Γ are said to lie in the same Tn -system if there exists some β ∈ Aut Γ such that T1 and β(T2 ) are Nielsen-equivalent. Thus a group Γ has only ﬁnitely many Tn systems if there exist ﬁnitely many epimorphisms φ1 , . . . , φm : Fn → Γ such that for any epimorphism φ : Fn → Γ there exists some α ∈ Aut Fn and some β ∈ Aut Γ and some i such that φ = β ◦ φi ◦ α. Thus the tree T in Theorem 1.1 can be chosen such that any vertex is in distance 1 from v0 iﬀ Γ has only ﬁnitely many Tn -systems. We will now show that for virtually free groups there can be inﬁnitely many T2 -systems. As all inner automorphisms lift to the free group it follows that a group that has inﬁnitely many Nielsen equivalence classes of generating n-tuples and has ﬁnite outer automorphism group must have inﬁnitely many T2 -systems. Thus the following theorem proves the claim. Theorem 6.1. There exists a virtually free group G with | Out(G)| < ∞ with a sequence (gi , hi ), i ∈ N, of generating pairs that are pairwise non-Nielsen equivalent.

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We ﬁrst record two lemmata. The ﬁrst one is due to Nielsen [N0] and is a reformulation of the well-known fact that any automorphism of the free group F (a, b) maps [a, b] to a conjugate of [a, b] or [a, b]−1 . It gives us a simple way to distinguish Nielsen equivalence classes of generating pairs. Lemma 6.2. Let G be a group and g1 , g2 , h1 , h2 ∈ G such that (g1 , h1 ) ∼ (g2 , h2 ). Then [g1 , h1 ] is conjugate to [g2 , h2 ] or [g2 , h2 ]−1 . The next lemma postulates the existence of a ﬁnite group with speciﬁc properties. Lemma 6.3. There exists a ﬁnite group H generated by conjugate elements x and y and such that there exists an element z ∈ H − x such that [x, z] = 1. Proof. is SL 2 (p) with odd p. SL2 (p) is generated

by thetwo matrices

An example 0 1 1 0 1 1 . The group which are conjugate by and y = x= 1 0 1 1 0 1 has further a non-trivial center which has trivial intersection with x. This proves the claim. Let now H be a group as in Lemma 6.3 and p be the order of x (and y). Choose w ∈ H such that wxw−1 = y. Choose further q such that p and q are ¯ = H ⊕ Zq , write Zq as additive group generated by 1. Note that coprime and let H the subgroup (x, 0), (idH , 1) is generated by (x, 1) and is cyclic of order pq. Let K = k | k 2pq be the cyclic group of order 2pq. We construct a graph of groups A as follows: 1. The underlying graph A has 2 vertices v1 and v2 with vertex group Av1 = K ¯ and v2 = H. 2. There exists an edge e with edge group Ae = a | apq , α(e) = v1 , ω(e) = v2 , αe (a) = k 2 and ωe (a) = (x, 1). 3. There exists a loop edge f with α(e) = ω(e) = v2 , edge group Af = H and αe (h) = ωe (h) = (h, 0) for all h ∈ H = Af . Let now G be the fundamental group of A. As all vertex groups of A are ﬁnite it follows that G is virtually free. Clearly G has the presentation

x, y, u, k, t | uq , [t, x], [t, y], [u, x], [u, y], k2 = xu, R where R is a set of deﬁning relators for H with respect to the generators x and y, t is the stable letter of the HNN-extension corresponding to the edge f and u corresponds to (idH , 1). Note that both u = k 2p and x = k 2q ; thus G is generated by k, y and t. We now put gi = k and hi = (kz)i tw−1 for all i ≥ 0 where z is the element postulated in Lemma 6.3. Together with the remark preceding Theorem 6.1 the proof of Theorem 6.1 is clearly an immediate consequence of the following three simple lemmata. Lemma 6.4. G = gi , hi for any i ≥ 0.

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Proof. Note ﬁrst that 2q −1 h−1 (kz)−i k 2q (kz)i tw−1 = wt−1 (kz)−i x(kz)i tw−1 = wxw−1 = y i gi hi = wt

as t, z and k commute with k 2q = x. Thus k, y ∈ gi , hi . As further hi = (kz)i tw−1 and w ∈ x, y ⊂ k, y this implies that t ∈ gi , hi . It follows that y, k, t ∈ gi , hi which proves the assertion of the lemma. Lemma 6.5. (gi , hi ) and (gi , hi ) are Nielsen equivalent iﬀ i = j. Proof. Let i = j. In the light of Lemma 6.2 we only need to show that [gi , hi ] is neither conjugate to [gj , hj ] nor to [gj , hj ]−1 . To do so it suﬃces to show that [gi , hi ] and [gj , hj ] act with diﬀerent translation length on the Bass-Serre tree of A or equivalently that the cyclically reduced reduced forms are of diﬀerent lengths. This is obvious as [gi , hi ] = k · (kz)i tw−1 · k −1 · wt−1 (z −1 k −1 )i can be represented by the cyclically reduced A-path k(k,e,z,e−1)i−1 ,k,e,z,f,w−1,e−1 ,k −1 ,e,w,f −1,(z −1 ,e−1 ,k−1 ,e)i−1 ,z −1,e−1 ,k −1 which is of length 4i + 4.

Lemma 6.6. The group G has ﬁnite outer automorphism group. Proof. By construction G is the fundamental group of a graph of ﬁnite groups with two edge groups Ae and Af such that neither is conjugate into the other. This guarantees the ﬁniteness of the outer automorphism group, see [P].

References [D]

M.J. Dunwoody Folding sequences Geometry & Topology Monographs. Volume 1: The Epstein Birthday Schrift, 139–158. [Del] T. Delzant, Sous-groupes a ` deux g´ en´erateurs des groupes hyperboliques. E. Ghys, A. Haeﬂiger and A. erjovski (ed.) et al., Group theory from a geometrical viewpoint. Singapore: World Scientiﬁc., 1991, 177–189. [HW] M. Heusener and R. Weidmann, Generating Pairs of 2-Bridge knot groups, to appear in Geom. Ded. [KW1] I. Kapovich and R. Weidmann, Freely indecomposable groups acting on hyperbolic spaces, IJAC 14 (2004), 115–171. [KW2] I. Kapovich and R. Weidmann, Kleinian groups and the rank problem, Geometry and Topology 9 (2005), 375–402. [KMW] I. Kapovich, A. Myasnikov and R. Weidmann. A-graphs, foldings and the induced splittings IJAC 15 no.1, 2005, 95–128. [KM] O. Kharlampovich and M. Myasnikov, Irreducible Aﬃne Varieties over a free group, J. Algebra 200, 1998, 517–570. [KPS] A. Karrass, A. Pietrowski and D. Solitar, Finite and inﬁnite cyclic extensions of free groups. Collection of articles dedicated to the memory of Hanna Neumann, IV. J. Austral. Math. Soc. 16, 1973, 458–466. [L] P.A. Linnell, On accessibility of groups J. Pure Appl. Algebra 30, 1983, 39–46.

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[W] [Z]

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A.I. Malcev, On isomorphic representations of inﬁnite groups by matrices Mat. Sb. 8, 1940, 405–422. J. Nielsen, Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden, Math. Ann. 78, 1917, 385–397. J. Nielsen, Om Regning med ikke kommutative Faktorer og dens Anvendlese i Gruppenteorien, Mat. Tidskrift B, 1921, 77–94. M.R. Pettet, Virtually free groups with ﬁnitely many outer automorphisms, Trans. AMS 349, no. 10, 1997, 4565–4587. A.A. Razborov, On systems of Equations in a Free Group, PhD thesis, Steklov Math. Inst., 1987. G.P. Scott The geometries of 3-manifolds Bull. Lond. Math. Soc. 15, 1983, 401– 487. Z. Sela, Diophantine geometry over groups I. Makanin Razborov Diagrams. Publ. Math. IHES 93, 2001, 31–105. J. Stallings, Groups of cohomological dimension one. Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVIII, New York, 1968), 1970, 124– 128. R. Weidmann, On accessibility of ﬁnitely generated groups, to appear in Q. J. Math. ¨ H. Zieschang, Uber die Nielsensche K¨ urzungsmethode in freien Produkten mit Amalgam, Inv. Math. 10, 4–37, 1970.

Richard Weidmann Department of Mathematics Heriot Watt-University Riccarton Edinburgh, EH14 4AS, UK e-mail: [email protected]

Combinatorial and Geometric Group Theory Trends in Mathematics, 307–315 c 2010 Springer Basel AG

Limits of Thompson’s Group F Roland Zarzycki −i i Abstract. Let F be the Thompson’s group x0 , x1 |[x0 x−1 1 , x0 x1 x0 ], i = 1, 2. −1 −i −1 i Let Gn = y1 , . . . , ym , x0 , x1 |[x0 x1 , x0 x1 x0 ], yj gj,n (x0 , x1 ), i = 1, 2, j ≤ m, where gj,n (x0 , x1 ) ∈ F , n ∈ N, be a family of groups isomorphic to F and marked by m + 2 elements. If the sequence (Gn )n<ω is convergent in the space of marked groups and G is the corresponding limit we say that G is an F -limit group. The paper is devoted to a description of F -limit groups.

Mathematics Subject Classiﬁcation (2000). 20E06, 20E18, 20F69. Keywords. Thompson’s Group F , limit groups, HNN-extensions, free products, group laws.

1. Preliminaries The notion of limit group was introduced by Z. Sela in his work on characterization of elementary equivalence of free groups [12]. This approach has been extended in the paper of C. Champetier and V. Guirardel [7], where the authors look at limit groups as limits of convergent sequences in a space of marked groups. They have given a description of Sela’s limit groups in these terms (with respect to the class of free groups). This approach has been also applied by L. Guyot and Y. Stalder [10] to the class of Baumslag-Solitar groups. Thompson’s group F has remained one of the most interesting objects in geometric group theory. We study F -limit groups. We show in this paper, that among F -limit groups there are no free products of F with any non-trivial group. Moreover, we prove that among F -limit groups there are no HNN-extensions over cyclic subgroups. In the remaining part of the section we recollect some useful deﬁnitions and facts concerning limit groups and Thompson’s group F . In Section 2 we present results concerning free products and in Section 3 results concerning HNN-extensions. A marked group (G, S) is a group G with a distinguished set of generators S = (s1 , s2 , . . . , sn ). For ﬁxed n, let Gn be the set of all n-generated groups marked by n generators (up to isomorphism of marked groups). Following [7] we put certain

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metric on Gn . We will say that two marked groups (G, S), (G , S ) ∈ Gn are at distance less or equal to e−R if they have exactly the same relations of length at most R. The set Gn equipped with this metric is a compact space [7]. Limit groups are simply limits of convergent sequences in this metric space. Deﬁnition 1.1. Let G be an n-generated group. A marked group in Gn is a G-limit group if it is a limit of marked groups each isomorphic to G. To introduce the Thompson’s group F we will follow [5]. Deﬁnition 1.2. Thompson’s group F is the group given by the following inﬁnite group presentation:

x0 , x1 , x2 , . . . |xj xi = xi xj+1 (i < j) In fact F is ﬁnitely presented: −i i F = x0 , x1 |[x0 x−1 1 , x0 x1 x0 ], i = 1, 2.

Every non-trivial element of F can be uniquely expressed in the normal form: 2 −a1 −a0 n xb00 xb11 xb22 . . . xbnn x−a . . . x−a x1 x0 , n 2

where n, a0 , . . . , an , b0 , . . . , bn are non-negative integers such that: i) exactly one of an and bn is nonzero; ii) if ak > 0 and bk > 0 for some integer k with 0 ≤ k < n, then ak+1 > 0 or bk+1 > 0. We study properties of F -limit groups. For this purpose let us consider a sequence, (gi,n )n<ω , 1 ≤ i ≤ t, of elements taken from the group F and the corresponding sequence of limit groups marked by t + 2 elements, Gn = (F, (x0 , x1 , g1,n , . . . , gt,n )), n ∈ N, where x0 and x1 are the standard generators of F . Assuming that such a sequence is convergent in the space of groups marked by t + 2 elements, denote by G = ( x0 , x1 , g1 , . . . , gt |RF ∪ RG , (x0 , x1 , g1 , . . . , gt )) the limit group formed in that manner; here x0 , x1 are “limits” of constant sequences (x0 )n<ω and (x0 )n<ω , gi is the “limit” of (gi,n )n<ω for 1 ≤ i ≤ t, RF and RG refer respectively to the set of standard relations taken from F and the set (possibly inﬁnite) of new relations. It has been shown in [7] that in the case of free groups some standard constructions can be obtained as limits of free groups. For example, it is possible to get Zk as a limit of Z and Fk as a limit of F2 . On the other hand, the direct product of F2 and Z can not be obtained as a limit group. HNN-extensions often occur in the class of limit groups (with respect to free groups). For example, the following groups are the limits of convergent sequences in the space of free groups marked by three elements: the free group of rank 3, the free abelian group of rank 3 or an HNN-extension over a cyclic subgroup of the free group of rank 2 ([6]). All non-exceptional surface groups form another broad class of interesting examples ([2], [3]). In the case of Thompson’s group the situation is not so clear. Since the centrum of F is trivial it is surely not possible to obtain any direct product with

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the whole group as an F -limit group. In 1985 Brin and Squier [4] showed that Thompson’s group F does not satisfy any law (also see Abert’s paper [1] for a shorter proof). However, in this paper we show that there are certain non-trivial words with constants over F (which will be called later laws with constants), which are equal to the identity for each evaluation in F . This implies that no free product of F with any non-trivial group is admissible as limit group with respect to F (see Section 2). Moreover, we prove that HNN-extensions over a cyclic subgroup are not admissible as limit groups with respect to F (see Section 3). There are many geometric interpretations of F , but here we will use the following one. Consider the set of all strictly increasing continuous piecewise-linear functions from the closed unit interval onto itself. Then the group F is realized by the set of all such functions, which are diﬀerentiable except at ﬁnitely many dyadic rational numbers and such that all slopes (derivatives) are integer powers of 2. The corresponding group operation is just the composition. For the further reference it will be useful to give an explicit form of the generators x0 , x1 , . . . in terms of piecewise-linear functions: ⎧ n ⎪ , t ∈ [0, 2 2−1 n ] ⎪ t ⎪ ⎪ ⎪ n n ⎪ −1 2n+1 −1 ⎨ 1 t + 2 n+1 , t ∈ [ 2 2−1 n , 2 2 2n+1 ] xn (t) = n+1 n+2 1 ⎪ ⎪ , t ∈ [ 2 2n+1−1 , 2 2n+2−1 ] t − 2n+2 ⎪ ⎪ ⎪ ⎪ n+2 ⎩ 2t − 1 , t ∈ [ 2 2n+2−1 , 1] for n = 0, 1, . . .. For any diadic subinterval [a, b] ⊂ [0, 1], let us consider the set, of elements in F , which are trivial on its complement, and denote it by F[a,b] . We know that it forms a subgroup of F , which is isomorphic to the whole group. Let us denote its standard inﬁnite set of generators by x[a,b],0 , x[a,b],1 , x[a,b],2 , . . .. Let us consider an arbitrary element g in F and treat it as a piecewiselinear homeomorphism of the interval [0, 1]. Let supp(g) be the set {x ∈ [0, 1] : g(x) = x} and supp(g) the topological closure of supp(g). We will call each point from the set Pg = (supp(g) \ supp(g)) ∩ Z[ 12 ] a dividing point of g. This set is obviously ﬁnite and thus we get a ﬁnite subdivision of [0, 1] of the form [0 = p0 , p1 ], [p1 , p2 ], . . . , [pn−1 , pn = 1] for some natural n. It is easy to see that g can be presented as g = g1 g2 . . . gn , where gi ∈ F[pi−1 ,pi ] for each i. Since g can act trivially on some of these subintervals, some of the elements g1 , . . . , gn may be trivial. We call the set of all non-trivial elements from {g1 , . . . , gn } the defragmentation of g. Fact 1.3 (Corollary 15.36 in [8], Proposition 3.2 in [11]). The centralizer of any element g ∈ F is the direct product of ﬁnitely many cyclic groups and ﬁnitely many groups isomorphic to F . Moreover if the element g ∈ F has the defragmentation g = g1 . . . gn , then some roots of the elements g1 , . . . , gn are the generators of cyclic components of the decomposition of the centralizer above. The components of this decomposition

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which are isomorphic to F are just the groups of the form F[a,b] , where [a, b] is one of the subintervals [pi−1 , pi ] ⊂ [0, 1], which are stabilized pointwise by g. Generally, if we interpret the elements of F as functions, the relations occurring −i i in the presentation of F , [x0 x−1 1 , x0 x1 x0 ] for i = 1, 2, have to assure, that two functions, which have mutually disjoint supports except of ﬁnitely many points, commute. In particular, these relations imply analogous relations for diﬀerent i > i 2. According to the fact that x−i 0 x1 x0 = xi+1 , we conclude that all the relations −1 of the form [x0 x1 , xM ], M > 1, hold in Thompson’s group F . We often refer to these geometrical observations. I am grateful to the referee for his helpful remarks.

2. Free products Brin and Squier have shown in [4] that the Thompson’s group F does not satisfy any group law. In this section we show how to construct words with constants from F , which are equal to the identity for any substitution in F . Deﬁnition 2.1. Let w(y1 , . . . , yt) be a non-trivial word over F , reduced in the group Ft ∗ F and containing at least one variable. We will call w a law with constants in F if for any g¯ = (g1 , . . . , gt) ∈ F t , the value w(¯ g ) is equal to 1F . The following proposition gives a construction of certain laws with constants in F . Proposition 2.2. Consider the standard action of Thompson’s group F on [0, 1]. Suppose we are given four pairwise disjoint closed diadic subintervals Ii = [pi , qi ] ⊂ [0, 1], 1 ≤ i ≤ 4, and assume that p1 < p2 < p3 < p4 . Then for any non-trivial h1 ∈ FI1 , h2 ∈ FI2 , h3 ∈ FI3 and h4 ∈ FI4 , the word w obtained from −1 −1 −1 −1 h1 yh4 , y −1 h−1 h2 yh3 ] [y −1 h−1 1 yh4 y 2 yh3 y

by reduction in Z ∗ F (we treat the variable y as a generator of Z) is a law with constants in F . −1 −1 h1 yh4 and w23 = Proof. We will use the following notation: w14 = y −1h−1 1 yh4 y −1 −1 −1 −1 y h2 yh3 y h2 yh3 . It is easy to see that w cannot be reduced to a constant. We claim that for any any g ∈ F satisfying g(q1 ) < p4 and g(p4 ) > q1 the word w14 (g) is equal to the identity. To show this we consider the action of w14 (g) on each point from [0, 1]. Assume, that t ∈ [0, g −1 (q1 )). Since t ∈ / supp(h4 ) we have: −1 −1 −1 −1 w14 (g)(t) = g −1 h−1 h1 g(h4 (t)) = g −1h−1 (h1 (g(t))). 1 gh4 g 1 gh4 g −1 (h1 (g(t)))) = By g −1 (h1 (g(t))) < g −1 (h1 (q1 )) = g −1 (q1 ) < p4 we see h−1 4 (g −1 g (h1 (g(t))). Thus: −1 (h1 (g(t))) = t. w14 (g)(t) = g −1 h−1 1 gg

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±1 If t ∈ [g −1 (q1 ), 1] then since h±1 4 (t) ≥ min(t, p4 ), we have g(h4 (t)) ≥ q1 and hence: −1 −1 w14 (g)(t) = g −1 h−1 h1 g(h4 (t)) 1 gh4 g −1 −1 = g −1 h−1 g(h4 (t)) = g −1 h−1 1 gh4 g 1 g(t) = t.

It follows that for g such that g(q1 ) < p4 and g(p4 ) > q1 , w14 (g) = 1F and hence w(g) = [w14 (g), w23 (g)] = [1F , w23 (g)] = 1F . Thus we are left with the case when g(q1 ) ≥ p4 (the proof of the case g(p4 ) ≤ q1 uses the same argument). Now we will prove that for any g ∈ F satisfying g(q1 ) ≥ p4 the word w23 (g) is equal to the identity. Assume that t ∈ [0, g −1 (p2 )]. Since g(q1 ) ≥ p4 , we have q1 ≥ g −1 (p4 ) > g −1 (p2 ) ≥ t. Thus: −1 −1 h2 gh3 (t) w23 (g)(t) = g −1 h−1 2 gh3 g −1 −1 −1 −1 h2 g(t) = g −1 h−1 g(t) = t. = g −1 h−1 2 gh3 g 2 gh3 g

Now assume that t ∈ (g −1 (p2 ), g −1 (q2 )). Then since again q1 ≥ g −1 (p4 ) > g −1 (q2 ) ≥ t we obtain: −1 −1 −1 −1 h2 gh3 (t) = g −1 h−1 h2 g(t). w23 (g)(t) = g −1 h−1 2 gh3 g 2 gh3 g

Since h2 (g(t)) ∈ (p2 , q2 ) we have g −1 (h2 (g(t))) < g −1 (q2 ) < q1 and: −1 −1 −1 h2 g(t)) = g −1 h−1 h2 g(t)) = t. w23 (g)(t) = g −1 h−1 2 gh3 (g 2 g(g

Assume that t ∈ [g −1 (q2 ), g −1 (p3 )]. Since we still have g −1 (p3 ) < q1 , we see that: −1 −1 w23 (g)(t) = g −1 h−1 h2 gh3 (t) 2 gh3 g −1 −1 −1 −1 = g −1 h−1 h2 g(t) = g −1 h−1 g(t) = t. 2 gh3 g 2 gh3 g

Let t ∈ (g −1 (p3 ), g −1 (q3 )). Then since g(p3 ) > q2 and h3 (t) = t ⇒ h3 (t) > p3 : −1 −1 w23 (g)(t) = g −1 h−1 h2 gh3 (t) 2 gh3 g −1 −1 = g −1 h−1 gh3 (t) = g −1 h−1 2 gh3 g 2 g(t) = t.

Finally assume t ∈ [g −1 (q3 ), 1] (and then g(t) > q2 ). Similarly as above: −1 −1 w23 (g)(t) = g −1 h−1 h2 gh3 (t) 2 gh3 g −1 −1 = g −1 h−1 g(h3 (t)) = g −1 h−1 2 gh3 g 2 g(t) = t.

Now we see that for g such that g(q1 ) ≥ p4 , we have w23 (g) = 1F and hence w(g) = [w14 (g), w23 (g)] = [w14 (g), 1F ] = 1F . The proof is ﬁnished. We now apply the construction from Proposition 2.2 to limits of Thompson’s group F . Theorem 2.3. Suppose we are given a convergent sequence of marked groups ((Gn , (x0 , x1 , gn,1 , . . . , gn,s )))n<ω , where Gn = F , (gn,1 , . . . , gn,s ) ∈ F , n ∈ N, and denote by G its limit. Then G = F ∗ G for any non-trivial G.

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Before the proof we formulate a general statement, which exposes the main point of our argument. Proposition 2.4. Let H = h1 , . . . , hm be a ﬁnitely generated torsion-free group, which satisﬁes a one variable law with constants and does not satisfy any law without constants. Let G be the limit of a convergent sequence of marked groups ((Gn , (h1 , . . . , hm , gn,1 , . . . , gn,t )))n<ω , where (gn,1 , . . . , gn,t ) ∈ H, Gn = H, n ∈ N. Then G = H ∗ K for any non-trivial K < G. Proof. It is clear that G is torsion-free. To obtain a contradiction suppose that G = H ∗ K, K = {1}, and G is marked by a tuple (h1 , . . . , hm , f1 , . . . , ft ). Let ¯ f¯) be an element of K \ {1} and let w(y) be a law with constants in H. f = u(h, ¯ gn,1 , . . . , gn,t )) = 1H for all n < ω. It follows from the deﬁnition Obviously w(u(h, ¯ f¯)) = 1G . Since w was chosen to be non-trivial, of an H-limit group that w(u(h, with constants from H and |f | = ∞, we obtain a contradiction with the fact that G is the free product of H and K. Proof of Theorem 2.3. It follows directly from Proposition 2.2, that there is some word w(y), which is a law with constants in F , and hence we just apply Proposition 2.4 for H = F , h1 = x0 and h2 = x1 .

3. HNN-extensions Now we proceed to discuss the case of HNN-extensions. For this purpose we consider a sequence of groups marked by three elements, (Gn )n<ω , and the corresponding limit group G = ( x0 , x1 , g|RF ∪ RG , (x0 , x1 , g)). The following theorem is the main result of the section. Theorem 3.1. Let (Gn )n<ω be a convergent sequence of groups, where Gn = (F, (x0 , x1 , gn )), and let G = ( x0 , x1 , g|RF ∪ RG , (x0 , x1 , g)) be its limit. Then G is not an HNN-extension of Thompson’s group F of the following form

x0 , x1 , g|RF , ghg −1 = h

for some h, h ∈ F.

In what follows we will need two easy technical lemmas: 1 −b0 n . . . x−b be its normal Lemma 3.2. Suppose g ∈ F and let xa0 0 xa1 1 . . . xann x−b n 1 x0 form. There is M ∈ N such that for all m > M :

where t =|

n

i=0 (ai

g −1 xm g = xm+t or gxm g −1 = xm+t , − bi ) |.

n xb00 xb11 . . . xbnn x−a n

n

i=0 (ai − bi ) ≥ 0. Then for suﬃciently large 1 −a0 1 −b0 n . . . x−a x0 xm xa0 0 xa1 1 . . . xann x−b . . . x−b n 1 1 x0

Proof. Consider the case when

m:

1 −b0 n = xb00 xb11 . . . xbnn xm+ ni=0 ai x−b . . . x−b = xm+ ni=0 (ai −bi ) . n 1 x0 n In the case when i=0 (ai − bi ) < 0 we consider the symmetric conjugation and apply the same argument.

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Lemma 3.3. Under the assumptions of Lemma 3.2 the numbers M and t deﬁned in that lemma, additionally satisfy the property that for all m > M and k > 0: g −k xm g k = xm+kt or g k xm g −k = xm+kt . Proof. If g −1xm g = xm+t holds (one of possible conclusions of Lemma 3.2) then M ≤ m + t and applying Lemma 3.2 k times we obtain the result. The case gxm g −1 = gm+t is similar. Proof of Theorem 3.1. First we prove the theorem in the case of centralized HNNextensions. Suppose that h = h = 1 in the formulation, i.e., the limit group has a relation of the form ghg −1 = h and denote by H the corresponding HNN-extension of Thompson’s group x0 , x1 , g|RF , ghg −1 = h. Assume that ghg −1 = h is satisﬁed in G. From the deﬁnition of a limit group it follows, that gn hgn−1 =F h for almost all n. Denote by C(h) the centralizer of h and by C1 ⊕ · · · ⊕ Cm its decomposition taken from Fact 1.3. As almost all gn commute with h, almost all gn have a decomposition of the form gn = gn,1 . . . gn,m , where gn,i ∈ Ci . As h = 1, at least one of the factors C1 ⊕ · · ·⊕ Cm is isomorphic to Z, say Ci0 . Denote by [a, b] the support of elements taken from the subgroup Ci0 . It follows from the construction of this decomposition, that h can only ﬁx ﬁnitely many points in [a, b]. Let us consider the sequence (gn,i0 )n<ω . Without loss of generality we may suppose that (gn,i0 )ω<∞ consists of powers of some element of F (which is a generator of Ci0 ). Consider the case when it has inﬁnitely many occurrences of the same element. If g occurs inﬁnitely many times in this sequence, then inﬁnitely many gn (g )−1 commute with x[a,b],0 , x[a,b],1 . That gives us a subsequence (gkn ,i0 )n<ω for which the relation [gkn (g )−1 , f ] holds for all n and for all f ∈ x[a,b],0 , x[a,b],1 . As x[a,b],0 , x[a,b],1 is isomorphic to Thompson’s group F , we will ﬁnd a word of the form g y −1 f −1 y(g )−1 f with f ∈ x[a,b],0 , x[a,b],1 , which is trivial for y = limn→∞ gkn in the limit group corresponding to the sequence (gkn ,i0 )n<ω and non-trivial for y = g in the group H. Indeed, it follows from Britton’s Lemma on irreducible words in an HNN-extension ([9], page 181), that the considered word can be reduced in H only if f −1 lies in the cyclic subgroup generated by h. But f ∈ x[a,b],0 , x[a,b],1 can be easily chosen outside h. Let us now assume that the sequence (gn,i0 )n<ω is not stabilizing. By the discussion from the end of Section 1 we see that for all m > 1, [x[a,b],0 x−1 [a,b],1 , x[a,b],m ] = 1.

(†)

On the other hand any gn,i0 is a power of some ﬁxed element from Ci0 . Thus we see by Lemma 3.3 that for M found for the generator of Ci0 as in Lemma 3.2: −1 x g = x[a,b],M+tn (∀n) gn,i 0 [a,b],M n,i0

or −1 = x[a,b],M+tn , tn ≥ 0. (∀n) gn,i0 x[a,b],M gn,i 0

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Substituting x[a,b],M+tn into (†) instead of x[a,b],m we have: −1 [x[a,b],0 x−1 [a,b],1 , gn,i0 x[a,b],M gn,i0 ] = 1

or −1 [x[a,b],0 x−1 [a,b],1 , gn,i0 x[a,b],M gn,i0 ] = 1.

Thus we see that one of the following relations holds for all n: −1 [x[a,b],0 x−1 [a,b],1 , gn x[a,b],M gn ] = 1

or −1 [x[a,b],0 x−1 [a,b],1 , gn x[a,b],M gn ] = 1.

Suppose that the ﬁrst relation holds for all n’s. Consider the corresponding word in the group H: −1 −1 −1 x[a,b],M gx[a,b],0 x−1 x[a,b],M g. w = x[a,b],1 x−1 [a,b],0 g [a,b],1 g

We claim that w = 1 in this HNN-extension. Once again, it follows from Britton’s Lemma on irreducible words in an HNN-extension, that we can reduce w if x[a,b],M is a power of h or x[a,b],0 x−1 [a,b],1 is a power of h. We know that x[a,b],0 , x[a,b],1 , . . . , x[a,b],m , . . . generate the group F[a,b] , which is isomorphic to F . From the properties of F we know that for diﬀerent m, m > M , x[a,b],m and x[a,b],m do not have a common root. Thus, possibly increasing the number M , we d can assume that x[a,b],M is not a power of h. If x[a,b],0 x−1 [a,b],1 = h for some integer d, then hd ﬁxes pointwise the segment [ 14 a + 34 b, b] ⊂ [a, b]. Hence h also ﬁxes some ﬁnal subinterval of [a,b]. This gives a contradiction as h was chosen to ﬁx only ﬁnitely many points in [a, b]. This ﬁnishes the case of centralized HNN-extensions. Generally, let us consider the situation, where in the limit group we have one relation of the form ghg −1 = h for some h, h ∈ F . By the construction of limit groups h = hf for some element f ∈ F . Indeed, if h and h are not conjugated in F , then there is no sequence (gn )n<ω in F with gn hgn−1 = h for almost all n. We now apply the argument above: let (f gn )n<ω be a sequence convergent to the element f g. It obviously commutes with h, so we can repeat step by step the proof above. That completes the proof.

References [1] M. Abert, Group laws and free subgroups in topological groups, Bull. London Math. Soc. 37 (2005), 525–534. [2] B. Baumslag, Residually free groups, Proc. London Math. Soc. (3), 17:402–418, 1967. [3] G. Baumslag, On generalised free products, Math. Z., 78:423–438, 1962. [4] M.G. Brin, C.C. Squier, Groups of piecewise linear homeomorphisms of the real line, Invent. Math. 79 (1985), 485–498. [5] J.W. Cannon, W.J. Floyd, W.R. Parry, Introductory notes on Richard Thompson’s groups, Enseign. Math. (2) 42 (1996), 215–256.

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[6] B. Fine, A.M. Gaglione, A. Myasnikov, G. Rosenberger, D. Spellman, A classiﬁcation of fully residually free groups of rank three or less, J. Algebra, 200 (2):571–605, 1998. [7] C. Champetier, V. Guirardel, Limit groups as limits of free groups: compactifying the set of free groups, Israel J. Math. 146 (2005), 1–76. [8] V. Guba, M. Sapir, Diagram groups, Memoirs of the American Mathematical Society, Volume 130, Number 620, November 1997. [9] R. Lyndon, P. Schupp, Combinatorial group theory, Springer, Berlin 1977. [10] L. Guyot, Y. Stalder, Limits of Baumslag-Solitar groups and other families of marked groups with parameters, eprint arXiv:math/0507236. [11] M. Kassabov, F. Matucci, The simultaneous conjugacy problem in Thompson’s group F, eprint arXiv:math/0607167. [12] Z. Sela, Diophantine geometry over groups I: Makanin-Razborov diagrams, Publications Math´ematiques de l’IHES 93(2001), 31–105. Roland Zarzycki Institute of Philosophy University of Wroclaw ul. Koszarowa 3 51-149 Wroclaw, Poland e-mail: [email protected]