Geometric Galois Actions, Volume 2: The Inverse Galois Problem, Moduli Spaces and Mapping Class Groups (London Mathematical Society Lecture Note Series)

(bijections of TTI(V,X)) given by (p(a, /3)(g) = a-g-f}-1. Therefore the sheaf

H°(tRplCl*vA[1])/(vdA[1]))

equipped with the connection dc is an analogue of the local system P —> X x X in algebraic geometry. This suggests the following definitions of the Betti and De Rham fundamental groups. We set nf(V,x)~SpecHo(p*-1(x,x);Q). If V is a smooth quasi-projective scheme over a field k of characteristic zero and x is afe-pointof V then we set:

n?R(V,x) := Spec H°DR(p' 2.2. 2.2.1.

^fax)).

Let V be such as in 1.1. We assume additionally that The inclusion of complexes (C-%A1(V)-^A1{V)

A AX(V) -> 0) *->

induces an isomorphism H° := H^BaviC-^A^V^A^V)

A Al{V) -> 0)) « H°DR{pm ~l{x,x)).

(The bar construction is chosen to be compatible with the inclusion into the total complex of the De Rham complex of pm ~ 1 (x, x). The isomorphism means that to calculate HpR(pm ~ 1 (x, x)) we need only Al{V) and AX{V) A A\V).)

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233

We recall that H(V) = (A^V))*, T[H(V)} is the tensor algebra on H{V), R(V) := (A^V) A A^V))* is contained in A2H(V) and Q(V) := T H v i ( )}/(R(V))Let us set

where / is the augmentation ideal of Q(V). Recall that Q(V)

:=U n

and P(V) C Q(V). 2.2.2. Notice that the symmetric algebra on Q(V)* is the algebra of polynomial functions on Q(V). Let us fix a basis u)\,..., un of A1 (V). Let Xi,..., Xn be the dual basis of H(V). Then T[H(V)] = C{XU..., Xn} is the polynomial algebra on non-commuting variables Xi,...,Xn. Let (X^ • ... • Xik)* denotes the linear form on T[H(V)} which to / in T[H(V)] associates its coefficient at the monomial X^ • ... • Xik. Let T* be the vector space generated by all {Xh -...-Xik)* including 1*. Observe that Q{V)* is a subspace of T*. We equipped T* with a shuffle multiplication and a Hopf algebra structure given b y ( X i , •...• X i p ) * • ( X i p + 1 .....

XipJq)* =

£

( X i a W •...•

Xia(p+q)Y

a€Sh(p,q)

and (Xh •... • XihY

-+ J2(Xh

. . . . . Xity

0 (Xil+1 . . . . . Xiky.

The shuffle

multiplication and the Hopf algebra structure on T* induce the analogous structures on Q(V)*. H° is also equipped with a shuffle multiplication and a Hopf algebra structure which on the form level are given by the same formulas. Lemma 2.2.3. There is an isomorphism of Hopf algebras

given i y w ^ ® . . . ® ujik -> (X^ • . . . •

Xik)*.

Proof. We identify the linear form (X^ • ... • Xik)* with the vector (X^ ® ... 0 Xik )* =Ljil®...®uJik. Observe that H° is a subspace of C 0 A1 (V) 0 (A^V) 0 ^ O O ) 0 (A1{V)®A1{V) O-A 1 ^)) © ••• = r*. Both algebras H° and Q(V)* have a natural grading. The part of H° in degree 2 is equal to ker(A 1 (^) 0 AX(V) -> A1^) A A x (y)). This is also the part of Q(F)* in degree 2. One checks that H° and Q(F)*are isomorphic in each degree.

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Let Sym(T*) be the symmetric algebra on T*. Let Sh(V) be the ideal in Sym(T*) generated by the shuffle product relations

(xiY -... • xipY - (xip+1 •... • xip+q)* =

2 ^ ( \ ( 1 ) • • • • • xia{p+q))*.


Then Sym,(T*) divided by Sh(V) is isomorphic to T* equipped with shuffle multiplication. 2.2.4. This implies that the symmetric algebra on Q(V)* divided by the ideal Sh(V) flQ(V)* is isomorphic to Q(F)*equipped with the shuffle multiplication. Let Alg (TT(V)) be the algebra of polynomial functions on n(V). It follows from a theorem of R. Ree (see [R] Theorem 2.5) that a formal power series / e P(V) is in the image of exp : ir(V) -> P(V) C Q(V) if and only if its coefficients (i.e. the coefficients of some lifting to C{{Xi,..., Xn}}) satisfy shuffle product relations. Therefore 2.2.2 and 2.2.4 imply that exp : TT(V) -> P(V) induces an isomorphism of Hopf algebras 2.2.5

R : Q(F)*-^ Alg(n(V)).

Combining 2.2.1, Lemma 2.2.3 and 2.2.5 we get: Lemma 2.3. isomorphic.

The Hopf algebras H^R{pm~1{x,x))

and Alg(7r(V)) are

Corollary 2.4. The group schemes TT^R(V^X) and Spec (Alg (7r(V))) are isomorphic. We denote by u : 7rf>R(V,x) —> Spec (Alg (TT(V))) the isomorphism induced by isomorphisms 2.2.1, 2.2.3 and 2.2.5. 2.5.

Let k be the field of complex numbers C. We shall assume that

In section 1 we considered the principal 7r(F)-bundle equipped with the integrable connection given by toy We shall relate it to the objects described in 2.1. The group ir(V) acts on Alg(7r(V)) on the left by the formula a) 9(f)(x):=f(x.g)

b) g(f)(x) := f(g-lx)

We form the associated vector bundle V x Alg (n(V)) ztVx

n(V) x Alg (n(V)) -+ V 7T(V)

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235

and we equip it with the connection induced by uy. We shall denote this connection by u'v in the case a) and by fujy m the case b). Let tyVA[i])/(vd±[i]) k e the algebraic De Rham complex of smooth relative differentials on VAM (i.e. on each VAW»). Let H% (respectively x ^ ° ) be the restriction of the vector bundle to X x {x} (resp. {x} x X.) Both vector bundles H°(tRplQ*,vA[l])/,vdA[1])) are equipped with the induced connection, which we also denote by dcOur main result in this section is the following theorem. Theorem 2.6. Let V be as in 2.1. Then the algebraic vector bundles equipped with connections (H°x,dc)

and

{VxAlg(x(V))->V,ui'v)

are isomorphic in the category of algebraic vector bundles equipped with algebraic integrable connections. Remark. (xW°,dc) and (V x A\g(n(V)) —> V/uy)

are also isomorphic.

Proof. We shall calculate horizontal sections of the connection dc on % := #°(£i?p*£l? yA[1])/ , vaA[1] A We recall the construction of dc- Following Katz and Oda (see [KO]) the filtration {F^}i on £lyn+2 is defined by i W n + 2 := image(il^ +2 ® (pn)*fiVxV -> ^

+ 2

).

The filtrations {F^i for n = 0,1,2,... give a filtration Fi of QyA[1]. The connection dc is the connecting homomorphism of the functor H°(tRpl( )) for the short exact sequence of complexes on VrAt1l 0 _> Fl/F2 _+ F°/F2 _>. F°/Fi ^ o. On Vn+2 we have t)

Observe that Oy 0 fi^>n ® O V C Let us assume that 5^ u;^ .. .®tc;im € ^(V)®771 is closed in the total complex of fi*(p* ~1(a:, x)). Its cohomology class c belongs to HpR(pm ~ 1 (x, x)), the fibre of W over (x, x) G V x V. We face two problems. We must extend c to a continuous section of % and we must show that this section is horizontal. Let T := Tot(n -> Ov ® Q*(Fn) O Oy) be the total complex of the bicomplex (p, g) -> Oy ®

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Zdzislaw Wojtkowiak

top(Vq) ® Oy. The element u^ ® ... ® u^m can be interpreted as a global continuous section of T. One verifies that the class of the element

k+l<m

is a horizontal section of Uan. (Uan := U ®ov O^1). Hence we get: 2.7. Let (a,/?) € 7Ti(V X V,(x,a;)). Then the representation of TTI(V X V, (x,x)) on the vector space HjjR(pm ~1(x,x);Q) is given by (a, /3) : u)ix (8)... ® vim ->

0
^ , . . . , a;ifc )

k+l<m

By 2.2.5 the Hopf algebra Q(V)* is isomorphic to Alg (7r(y)). Hence we shall consider u>fv on the associated bundle V x Q(F)* —>• V. It follows from Proposition 1.9.i) that the monodromy representation is given by mi^x)

3 a : (Xt, •... • Xik)* -+ ^ ( X , , • . . . • Xit)

for the connection u'v, and by

for the connection 'uy. The isomorphism H°^>Q(V)* denned by wix ®.. . gives an isomorphism of monodromy representations of holomorphic vector bundles (H°x,dc)an and (V x Q(V)* -> F,o;^) an (resp. (xn°,dc)an and (V x Q(F)* -> V/a;v) an ) at the point xeV. (( ) an ) is an analytic object corresponding to an algebraic one ). Observe that both connections are regular. One can check this directly for the connection u'v. The connection dH(F,x)(C). Let a G TTI(V^X). There is a canonical map ni(V,x) —>

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237

TT^R(V,

x)(C) (the monodromy homomorphism) and it follows from 2.7 that the value of c on the image of a is equal to ^2 fa_1 u^,..., uJim. §3. Homotopy relative tangential base points on P 1 (C)\{ai, • • •, a n +i}. 3.1. In this section we define a tangential base point on the complex projective line minus a finite number of points. Definition 3.1.1. Let X = P1(C) \ {ax, ...,a n + i}. A tangential base point on X is a non-zero tangent vector v at one of the missing points a*. We shall denote by v (or by v) the tangential base point defined by the tangent vector v. If X = C\ {ai,..., an} we identify the tangent space at a^ with the vector space C and the tangent vector v at a* with the vector aix , where x = di+v. The complex projective line P 1 (C) has two canonical charts C —> P 1 (C), z -> [z : 1] and C —> PX(C), z —> [1 : z] which allow to identify tangent vectors to P 1 (C) with tangent vectors to C. Let A; be a subfield of C. We say that a tangential base point v at a^ is a tangential base fc-point if ai G k and a^ 4- v G k. Definition 3.1.2. Let X = PX(C) \ {au ...,a n +i} and let v be a tangential base point (at the missing point a*). A path on X starting from the tangential base point vis a smooth path in PX(C), satisfying the following conditions: i) the image of the open interval (0,1) is contained in X; ii) the tangent vector to the path at a^ is equal to v. We say that a path 7 ends at a tangential base point v if the path 7" 1 (7 -1 (£) := 7(1 — t)) starts from the tangential base point v. If a path 7 on P 1 (C) starts from v or ends at w or starts from v and ends at w and if 7((0,1)) C X then we say that 7 is a path on X. Definition 3.1.3. A homotopy between two paths 7 and 7' on X starting from a tangential base point v is a smooth homotopy H8 between the paths 7 and 7' on PX(C) satisfying the following conditions: i) # s ( ( 0 , l ) ) c X f o r s G [ 0 , l ] ; ii) the tangent vector of the path Hs at the starting point is v for s G [0,1], A homotopy between two paths 7 and 7' which end at a tangential base point v is a homotopy between the paths 7" 1 and j ' ~ l which start at v. According to this definition, the paths a, f3 on C \ {0} shown in the following figure are not homotopic.

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Zdzislaw Wojtkowiak

w Definition 3.1.4. Let 7 be a path starting from a tangential base point v (a tangent vector at a^) and ending at the same tangential base point v. We say that 7 is null homotopic if there is a smooth homotopy Hs between 7 and the constant map into a^ such that: i) Hs is a path on X starting from v and ending at v for s < 1; ii) Hi is the constant map into a^. If 7 is a path on X we denote by [7] the homotopy class of 7. Suppose that a path (p on X ends at a tangential base point v and a path ip on X starts from v. We define a composition of homotopy classes [ip] • [ 0 be small. Let (fi-£ := y?|[o,i-e] and ^ e := ^|[e,i]- We can assume that <^(1 — e) = ip{e). We define the composition of paths ip • e • ^ i _ e (with a necessary reparametrization and smoothing). While i\) •
n+1

Let X = X U |J (Ta. (P1(C))\{0}) be a sum of X and the tangent spaces i=l

minus zero at the missing points. We define a groupoid P over J x X i n the following way. If (x, y) G X x X then P^y is the set of homotopy classes of paths on X starting from x and ending at y. We set P = U,x y)exxx^^y We define a projection p : P -> X x X by p(Px,3/) = (#, 2/)- The partial composition of homotopy classes makes p : P —> X x X into a groupoid over X x X. The fibre over (x,x), TT^X^X) := p~1(x,x) is the fundamental group of X at a base point x G X. Observe that x can be a tangential base point v. 3.2. We shall construct functions Ax{z; v, 7) if v is a tangential base point. Let us set X = C\{ai,..., an}. Let XQ G C and let # : [0,1] 3 t —>> a; + t • (#0 ~~ a i) be an interval joining ai and XQ. Let 7 be a path from a^ to z £ X (not passing through any a^, fc = 1,.. .71) tangent to 5 in a*. We assume

Monodromy of Iterated Integrals

239

that in a small neighbourhood of a* the path 7 coincides with 5. We identify v — XQ - cii with a tangent vector to C in a^. Let u)\ — ^z^7? • • • > ^n — ztzan • We set

if ax / L J a,i

Let e £ im(S) be near a*. Let 7^ be a part of 7 from e to 2, and let S£ be a part of 6 from £ to :co- We set

-\, \

r

dt

f

:

A

= }™ I

dt

a i i tr(*,...,*)(*) w

and

:= / Jatn

Aa $(i,.. .,i,ak+i..

.,ai)(t)ujaa/+l

if a^-f 1 7^ z, and / > k + 1.

L e m m a 3.2.1. T/ie integrals Aaii#(aii /2/^'c, multivalued functions on X.

• • • ^fcX-z) exis^ and t/iey are ana-

Proof. Assume that at ^ i for t < I and ai+i = i. The function g(z) := f*. (jjai,..., uJai is analytic multivalued on X U {a^} and vanishes in a^. Hence the integral g\{z) := J^. ^(^)^z^: exists, the function g\(z) is analytic multivalued on X U {a;} and vanishes in a^ Hence by induction we get that A a i ,v(^i,..., an)(jz) exists and it is analytic, multivalued on X U {ai}. Assume now that at — i for t < I and a/+i 7^ i. Without loss of generality we can assume that a^ = 0 and XQ = 1. Observe that lim

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Zdzislaw Wojtkowiak

where /% are rational numbers. The function zq(logz)p for q and p positive integers, is analytic multivalued on X, continuous on any small cone with a vertex in a; (0 in this case) and it vanishes in a^. The function j ^ - for j 7^ i is bounded on any sufficiently small neighbourhood of a*. Hence the integral lim im

f

is an analytic multivalued function on V, continuous and univalued on any small cone with a vertex in a* and it vanishes in a^. This also implies that the last integral exists. 0 We recall that X x , . . . , Xn are duals of 7^7, • • • > jz^-

Let us set

AxOz; ff,7) = 1 + $^(-l)*A o < | i r (ai,...,a*) (*)*<,, •... • Xai. Observe that Ax (2; #,7) depends on v, i.e. on XQ. Lemma 3.2.2. The map X 3 z -> (z,Ax(z;v,j))

e X x P(X)

is horizontal with respect to UJX • Proof. We have dA . a

Hence the functions (—l)fcAa<)j?(ai, • • • »«fc)(^) satisfy the system of differential equations defining horizontal sections of the principal bundle X x 0 P(X) —> X equipped with the connection LUXIt remains to define functions Ax(z; #,7) if a^ = 00 or all ai are different from 00. Let / : Y = C\{6i,... ,bn} -> X = P x (C)\{ai,... , a n + i } be a regular map of the form ^ ^ with det (

J ^ 0. Let y G Y and let

7 be a path on Y from 2/ to z. Then it follows from corollary 1.7 that ifyeY. We shall use this fact to /*(A y (z;y,7)) = Ax(f(zyj(y)J(j)) define Ax (z\ v, 7) where v is a tangent vector to P 1 (C) in a^ and 7 is a path from a,i to z, which is tangent to v. We set (3.2.2.1.)

A^(z;t;,7) := /•(Ay(/- 1 (z);/,- 1 (fl),/

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241

It is clear that Ax(z;v^) does not depend on the choice of / . Thus we have the following lemma. Lemma 3.2.3. Let X = P1(C) \ {ai,... ,a n +i} and let v be a tangential base point. Then the map

is horizontal with respect to UJX • We set Lx{z\ v, 7) := \ogAx{z\ v, 7). If we are dealing with only one space X we shall usually omit the subscript X and we shall write A(z;v, 7) and L(z;v,7), or A^(z;y) and L#(z;7), or even A${z) and L${z). The constructions from 3.1 and 3.2 are summarized in the following proposition. Proposition 3.2.4. Let v G X. The functions Ax{z\v,'y) depend only on the homotopy class ofj in Pvz .

and Lx(z;v^y)

§4. Generators of 7Ti(P 1 (C)\{ai,... ,a n + i},x). Let X = P 1 (C)\{ai,..., a n + i} and let x G X. Let vl be a tangent vector in a&. Then the loop around a& at the base point v*k is the following element 5^. of?ri(X, v*k) (see picture).

a i)

ii)

iii)

Now we shall describe how to choose generators of TTI (X, x). Let us choose tangent vector v{ G Tai (P^Q) \ {0} for each i = l , 2 , . . . , n + l . If x G X then we choose a family of paths F = {7i}"J"11 on X from x to each ^ such that any two paths do not intersect and no path self intersects. The indices are chosen in such a way that when we make a small circle around x in the opposite clockwise direction, starting from 71, we meet 72,73,..., 7n+iIf x = v{ then we choose a family of paths F = {7i}^J21 fr°m x t o e a c h Vi such that making a small circle around a\ in the opposite clockwise direction, starting from 72 we meet 73,... ,7 n +i. We associate the following element Si in TTI(X, X) to the path 7*. We move along 7i, we make a small loop in the opposite clockwise direction around

242

Zdzislaw Wojtkowiak

di and we return to x along 7" 1 . In the case ii) the element S\ is the loop

sv-x. iv) We can choose any order of missing points to construct the elements The following lemma is obvious. Lemma 4.1. The elements Si, S2, • • • > ^n+i are generators o/7Ti(X, X). We have 5 n +i •... • S2 * S\ — 1. Definition 4.2. The ordered sequence ( £ 1 , . . . , S^+i) of elements of TTI(X, X) obtained from the family of paths F as in i),...,iv) will be called a sequence of geometric generators of TTI(X, X) associated to F. §5. Monodromy of iterated integrals on P 1 (C)\{ai,..., a n +i} Let X = P 1 (C)\{ai,..., a n +i}. We want to compare functions Av(z) for different base points. Let £1, £2, #3 6 X and let ZQ € X. Let Si for i = 1,2,3 be a path on X from Xi to z$. Let us set 7^ := 5" 1 o J^. Proposition 5.1. Let us continue each function AXi(z) along Si to the point ZQ. There exist elements 0^.(7*^) G P(X) such that

for all z in a small neighbourhood of z0. The elements a%i.(/yij) satisfy the following relations

(Observe that the elements a^. (7^) depend on the choice of paths 7i?.) Proof. The existence of a** (7^) follows from the fact that the AXi(z)'s are horizontal sections. The fir^t two relations are obvious. The last relation follows from the equalities Ax.(z) • a** (7^) = AXj(z), AXj(z) • axi(
where Av(z) and Aw(z) are continued along 7.

0

Monodromy of Iterated Integrals

243

Proposition 5.2. Let v*k G TafcP1(C)\{0}. Let Sv-k be a loop around ak based at v*k G Tak (P1(C))\{0} (see figure at the beginning of section 4)- The monodromy of Av-k (z) along Sv-*k is given by

Proof. The monodromy of Av-k(km)(z) := Av~(fc, &,..., k)(z) along S^ is given by Sv-*k : A,* (*")(*) -> Av-k(km)(z) + £ K-h(km-l)(z)l=%±.

Let

7 be a path on X from v*k to z and let ai ^ fc. We must analytically continue the function Av-k (fcn, a i , . . . , ap)(z) along the composition 7 • 5 ^ . The contribution along Sv-k can be made arbitrarily small. However the function Av-k(km)(z) changes after the tour along Sv-k. This implies that the monodromy of Av-k (A:m, a i , . . . , ap)(z) along Sv-i is given by

(

Av^(fc

o

-\l

,

1=1

for a\ / k. The formula for the monodromy of Ay*k (z) along 5 ^ follows from this. 0 Let x G X. Let us choose v{ G T^P^QXJO} for i = 1,2,..., n + 1. Let ( 5 i , . . . , Sn+i) be a sequence of geometric generators of TTI(X, X) associated to F = {ji}^i (if x G X) or F = {7i}^ 2 1 (^ x *s a tangential base point, i.e. x = tJi), where F is a family of paths on X from x to t^ for i = 1,2,..., n + 1 or i = 2 , . . . , n + 1 as in section 4. Theorem 5.3. The monodromy of the function Ax(z) along the loop Sk is given by Sk : Ax{z) -»• Ax(z) • (al*^))'1

• e-2«iX* •

af{lk).

Proof. It follows from Proposition 5.1 that for z in the small neighbourhood of some 7fc(£). This equality is preserved after the monodromy transformation along S&, hence we have (*2)

{Ax(z))s» = {kv-k{z))s» •

af{lk),

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Zdzislaw Wojtkowiak

where ( )Sk denotes the function ( ) after the monodromy transformation along Sy>k. Observe that (Av-k(z))Sk = (Av-k(z))Strk. It follows from Proposition 5.2 that (Av-k(z))s*=Av-h(z).e-2*iX*.

(*3)

Substituting (*3) in (*2) and then substituting (*i) for Av~k(z) yields the formula for (Ax(z))Sk. 0 Corollary 5.4. The monodromy of the function Lx(z) along the loop Sk is given by

Sk : Lx{z) -> Lx(z) • KHTfc))"1 • ( where af{^k) = \og{af{^k)). Proof. The corollary follows immediately from Proposition 1.8. ii).

0

Definition 5.5. Let x € X. Define a homomorphism ex,x : TTipT,*) -> P(X) (resp.^, x : 7n(X,x) -> n(X)) by the formula 1

*iX

af (7fc) (resp.

9XtX(Sk)

it is called the monodromy homomorphism of the form ux at the base point x. Observe that x can be a tangential base point. If x € X then the homomorphisms from Definitions 1.3 and 5.5 coincide. Let v,v' £ X and let 7 be a path on X from v to v'. If TTI(X, V) and 7Ti(X, v') are identified via the path 7, then the homomorphisms 6v,x and #v',x are conjugate. Proposition 5.6. Let

be a regular bijective map. Then for any v,w and any path S from v to w we have

G X , any path 7 from v to z

UAx(z;v,'Y))=AY(f(z);f(v)J(1)) and

Monodromy of Iterated Integrals

245

(notation: f(v) := f*(v) if v is a tangent vector). Proof. The proposition follows from the definitions of Kx{z\v^) for tangent vectors, and from Corollary 1.7.

and a^ 0

§6. Calculations. Let V = P 1 (C)\{0,1, oo}. The forms & and ^ form a basis of Ax{y). Let X := (&y and Y := ( ^ ) * be the dual basis of (A^V))*. Set Z := —X — Y. The group P(V) is the group of invertible power series in noncommuting variables X and Y with constant term equal to 1. Fix a path 71 = interval [0,1] from 01 to 10. It follows from Proposition 5.1 that along the path 71 we have

Let f(z) = l — z. It follows from Proposition 5.6 that

(We omit the arrow over vectors, when it does not cause confusion.) Proposition 5.1 implies that

Observe that /*(X) = Y and f*(Y) = X. Hence we get Deligne's formula (2)

al\{X,Y).al\{Y,X) = \.

(The proof of (2) given here essentially repeats that of Deligne (see[D5]).) Let us fix a path 7 ^ = interval [0, e] 4- arc from e to — e passing through (-i) • e + interval [-e, 00] from Ol to 00O where e > 0 (see figure). Yl

Let So (around 0), Si (around 1) and SQQ (around 00) be geometric generators of TTI(V, 01) associated to the family {71,700} ( s e e picture on page 5). Then we have So - Si - 5 ^ = 1.

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Zdzislaw Wojtkowiak

Theorem 6.1. The monodromy of A^z) las:

is given by the following formu-

So : A ol (z) -> A ol (z) • e (3) 5i : A ol (z) -> Aol(*) • « ( X , r ) ) " 1 • e^2^Y • aJ?(X, F), Soo : A ol (z) -> A ol (z) • e~«iX • « ( X , Z))" 1 • e<-a*«>z • < ( X , Z) Proof. The formulas for the monodromy along So and Si follows from Theorem 5.3. The monodromy along S ^ needs some explanation. By Theorem 5.3 it is given by the formula 5oo : A ol (s) -> A o l ( z ) > ^ ° ( X , Y))- 1 -^- 2 ™^a%?(X,Y). By Proposition 5.1 a^°(X,Y) = a%£(X,Y) • a$°(X,Y). One calculates using 3.2.3 that a(Jf(X,F) = eniX. Let /(z) = ^ f j . We have = X and /*(y) = Z. It follows from Proposition 5.6 that oJ?(X, Z) = Hence we get the formula describing the monodromy along SQQ.

§

We recall that the Lie algebra L(V) is the completion of the free Lie algebra over C on two generators X and Y. The group n(V) is L(V) equipped with the multiplication given by the Baker-Campbell-Hausdorff formula. Let us set a(X,Y) := a$(X,Y) := log(oJ2(X,y)). It follows from Corollary 5.4 that the monodromy of L o l( z ) ^s gi v e n by the following formulas: (4) Si : L ol (z) -> L ol (z) • a(X, F ) " 1 • (-2?ri)y • c*(X, y), oo : LQ^(Z)

—>• ^ o i v / * v ^ v ^ - ' ^ v ^ ^ A)

' \—*'R%)Zi - o:(A, Z) • (7^^jA.

Observe that Theorem A (in the introduction) follows from the formulas (4). Let us compute the coefficients of a^(X,Y) and a(X,Y). If w is a monomial in X and Y we denote by a(w) the coefficient of agi(X, Y) at w. Let X be the first Lie basis element and let Y be the second element. We choose a basis of a free Lie algebra on X and Y as in [MKS] pages 324-325. If w is an element of this basis, let a(w) be the coefficient of w of a(X, Y). Observe that the coefficients a(Xn) and a(Yn) vanish. We assume that z approaches 1 in the formula (0). Then it follows from the formula (0), the formula (1) and the formula (2) that (5) a(XnY) = (-l)"C(n + 1),

a(YnX) = (-l) n + 1 C(n + 1)

Monodromy of Iterated Integrals

247

and a(XiYj)+a(YiXj) = 0. (If a; is a one-form then UJ1 := O;,CJ, ...,u> i-times.) Let us set ctij := a((y, X ) ^ * - 1 ^ - 1 ) , where t i ^ y * - ((... ((w, X)X).. )X)Y).. . ^ . C o m paring the coefficients at Y^X1 and X*y^' in a(X, Y) and in log(aJ;J(X, we get It follows from the equality a^Y^)+0,^X3) = 0 that aitj = ajyi. Observe { j j that we also have a{X Y ) + (-1)*+*a(Y X*) = 0. This last equality can also be obtained from formula (1.6.2) in [Chi]. Let us set nf := (ir(V),n(V)), *" := (TT',^) and TT2(V) := ^(V)/^ 7 . It follows from (3) that the monodromy homomorphism 6^ : TTI(V,01) —> is given by

(7)

SQ ->(Si -^(-27rz)y + [-2?rzy, a(X, Y)]

= (-2ni)Y+ £ (27ri)a i=O,j=O

Let us set F{z) =

(los(

^ z ) ) "'. The formula

(which can be proved by induction) implies that

(8)

a n + h m ={

L)

y>

71.TH.

6.2. Suppose that rti > 1 for i = 1,..., k — 1 and n^ > 2. Then we have oo

These numbers are values of the functions

)Sk

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Zdzislaw Wojtkowiak

at z = 1 and ( s i , . . . , sk) = (n x ,..., nk). §7. Configuration spaces. and V = P ^ Q Y K , . . . , < + 1 } . If the Let V = V1(Q\{au...,an+1} sequences (a,x) := ( a i , . . . ,a n + i,x) and (a^x') := (a^,... ,a^ + 1 ,x / ) are close then the groups TTI(V, X) and TTI(V, a;') are canonically isomorphic. We shall study how the monodromy homomorphisms 9XiQr := 6xy &nd #x',a' : = 9x>y> from sections 1 and 5

depend on a and a'. If T is a topological space we set T? = {(tu . . . , £„) G T n | U ^ ^ if i ^ j } . Let Wn := C™. The space A1(TVn) of global holomorphic one-forms on Wn with logarithmic singularities is spanned by ujij — d ^~^ Z j for i,j e { 1 , 2 , . . . , n} and i < j . Let Xy = ( d ^ l f i ) * Xji = Xij and X« = 0. In A2(Wn) we have

b e their

formal duals. We set

Vki A CJ^ 4- cjjfc A LUki H- cJij A u ; ^ = 0

for z, j , A; different. These are the only relations between the 2-forms ujij (this result is due to V.I.Arnold). Dualizing the map A\A1(Wn))-+A1{Wn)AA1(Wn) we find that R(Wn) is generated by [Xij,Xik

4- Xjk] with i,j,k

different

and [Xij^Xki] with i , j , k,l different (see [C]). Let x = ( x i , . . . , x n , x n + i ) G W n +i be a base point. Let pi : W n + i -> Wn (i = 1 , . . . , n -h 1) be a projection p i ( z i , . . . , z n + 1 ) = ( z 1 ? . . . , zh . . . let

and let F(i, x) := p " 1 (x(i)) =

Monodromy of Iterated Integrals

249

(z means z is omitted). Let ki : V(i,x) —>> W n + i be given by ki(z) = (#1,..., Xi-i, z, Xi+i,..., x n +i). The inclusion ki induces

and

One calculates that (ki)*

/dzi-dzj\ \

Zi — Zj /

=

dz Z — Xj

and (ki)*

(dzi-dzj\ \ Z\ — Zj J

. , . , .

= 0 if I ^ % and j / i.

This implies that (ki)*(Xj) = Xij for j — 1,2,..., % - 1, i + 1,..., n + 1 where X/ is the formal dual of jz^~- It follows from the form of the generators of R(Wn) that the Lie subalgebra of Lie(Wn) generated by X^i,.. .Xij-i, Xiyi+i,... ,X i ? n + i is free and it is a Lie ideal of Lie(Wn). Hence the map (/?»)* is injective and its image, (ki)*(n(V(i, x))) is a normal subgroup of 7r(Wn+i). Let x = (a?i,...,a: n ,x n+ i) G Wn+i and x' = (xi,... , ^ , x ^ + 1 ) G W n +i. Let us set V := V(n + l,x) and V =: V(n + l,x'). We choose a family of non-intersecting paths 7 1 , . . . , 7 n , 7n+i in C from #1 to x[,..., xn to x^ and xn_|_i to x'nJtl. We shall identify 7Ti(V,a;n+i) and Ki(V,xfn+1) in the following way. Observe that 7 = (71,..., 7 n , 7 n +i) is a path in Wn+i from x to x'. The identification isomorphism 7. : 7ri(F,a;n+i) —> 7i"i(^',^n+i) i s the unique isomorphism making the following diagram commute ^

7Ti(W n+ i,x)

i 7#

is induced by the path 7 in a standard way). Proposition 7.1. After the identification of the fundamental groups of V = C\{xi,... ,xn} and V = C^x^,.. .,xfn} via 7, the monodromy homomorphisms 0Xu+ltv ' *i(V,xn+1)

-> ir(V) and 0x>n+1,v : ^i(V',x'n+1)

-+ n(V) = n(V)

are conjugated by an element of the group 7r(Wn+i). (The group (kn+i)*n(V) is a normal subgroup ofTt(Wn+i) so 7r(Wn-|_i) acts on K{V) by conjugation.)

250

Zdzislaw Wojtkowiak

Proof. The result is a consequence of the following commutative diagram: * ^ £

TT(V)

where c ^ (a;';a;,7) 1S a conjugation by the element Lwn+I(xf]x,j). It follows from Proposition 1.4 that the square (1) commutes. Corollary 1.7 implies that (2) and (3) commutes. The square (4) commutes by the construction. 0 Corollary 7.2. Let x = (xi,.. .,x n +i) € Wn+i. Se£ V(i) := V(i,x). Let dij be a geometric generator ofni(V(i),Xi), which is a loop around the point Xj. Let Aij be its image in 7Ti(Wn+i,a;). Then 0x,wn+1(Aij) is conjugate to (—27ri)Xij in the group 7r(Wn+i). Proof. It follows from Theorem 5.3 that ^Xi,y(i)(^ij) is conjugated to (—27ri)Xij in the group n(V(i)). Hence the corollary follows from Corollary 1.7. 0 Now we shall study the relation between the monodromy representations for the configuration spaces (P 1 (C)\{0, l,oo})? and ( P ^ C ^ O , l,oo})™. We shall use the Ihara result (see [II] The Injectivity Theorem (i)). Let us set Yn = (P^C))" and yn = (P^qXfO, l,oo})^" 3 . The group PGL2{C) acts diagonally on Yn and yn = Yn/PGL2(C). The identification is given by the map (0,1, oo, xux2,..., x n _ 3 ) -> (xi, x2,..., x n _ 3 ). Let ^ : W n _i -> 3^n be the composition of the map ( x i , . . . , Xfc_i, Xfc+i,..., x n ) —> (xi,.. .,Xfc_i,oo,Xfc+i,.. .,x n ) and the projection Yn —t yn. The map ipk induces (fc)* : ff(W n _i) -> H(yn). Let us set X^- = {i/>k)*{xij) where X^ = ( ^.Z^ )* ^ ^(W^n-i)- (We use the same notation for Xij G if(W n _i) and its image in H(yn). Notice also that it can be shown by computation that Xij in H(yn) does not depend on the choice of ipk-)

Monodromy of Iterated Integrals

251

Let Aij € fl"i(Wn_i, x) be as in Corollary 7.2. We shall also denote the image of A^ in yn by A^. Corollary 7.3. The element 0yyn {Aij) is conjugate to {—2ni)Xij in 7r{yn). Proof. This follows from Corollary 7.2 and the commutative diagram ^x)

->

7T(W n _i)

0 Let Aut(7r(3^n)) be the group of algebraic automorphisms of a complex pro-algebraic group 7r(yn). Let Aut*(7r(3^n)) be a subgroup of Aut(7r(3^n)) denned in the following way: Aut*(7r(yn)) = {/ E Aut(7r(y n ))|3a/ € a (~ means "is conjugate to".) Let us set Tn{Q ={ipe Hom(7ri(;yn,y);7r(;yn))|3av € C* {A^ E 7Ti(yn,2/) are as in Corollary 7.3.). The group Aut(7r(Xi)) acts freely on Tn{C). We indicate arguments that the action is transitive. The homomorphism tp £ Tn{C) factors through the Malcev C-completion ofni{yn, y) and it becomes an isomorphism Out*(7r(J;n_i)) is injective for n > 5. Proof. Let Outl(7r(;yn)) := ker(Out*(7r(yn)) 4 C ) , where N{f) = af. The Lie algebra of Out*(7r(3^n)) is the Lie algebra of special derivations of L{yn) modulo inner derivations. The Lie version is proved in [II] section 4. The categories of complex unipotent algebraic groups and nilpotent Lie algebras of finite dimension over C are equivalent. The group is pro-unipotent. Hence the Lie version implies the result for and then also for Out*{7r{yn)).

252

Zdzislaw Wojtkowiak

The surjective homomorphisms (pn_j_i)* : /K\(ynjti^y) —> K\(yn,y') {Pn+i)* : ^(Xi+i) —t ^(iVn) induce a morphism of torsors

and

compatible with Out*(7r(3^n+1)) -> Out*(7r(yn)). Lemma 7.5. T/ie canonical morphism of torsors tn+1(C) —>• £n(C) is mjective for n > 4. This follows immediately from Proposition 7.4. Corollary 7.6. Tfte monodromy homomorphism 0yyn : 7Ti(^n,2/) —»Tr^n) zs determined (up to conjugacy by an element of 7r(yn)) by the homomorphism 9yiyA : 7riCy4,2/') -^ TTQ^). Proof. Observe that ^,3; n G t n (C) and ^',^4 is the image of 6yyn under the canonical morphism £n(C) -> £4(C). Let a := ( a i , . . . , a n , a n + i ) be a sequence of n + 1 different points in P 1 (C) and let Va := P 1 (C)\{ai,..., a n , a n + i } . The vector space H(Va) is spanned

by Xi := ( ^ - - i z t r r ) * * = 1,.. •, n. Let us set Xn+1 := - E? = 1 Xt. Let Afc denote a geometric generator of Va, which is a loop around a&. Let us set Ta(C) := {/ € Hom(7n(Vra,x);7r(Vra))|3a/ G C*,V^lfc

f(Ak)

Assume that a = (a^)^ 1 is such that ai = 0, a2 = 1, as = 00. The fibration Va —

>yn+2 —

>yn+l

(Va =

1

realizes ^(Va) as a normal subgroup of ^(^+2) ((fcn+2)*(Xj) = Hence the group 7r(3^n_|_2) acts on Ta(C); let U Q := T a (C)/7r(^ n+ 2). Observe that any 7r(J;n+2)-conjugate of Xi>n+2 is in the image of ir(Va). Hence the restriction map (kn+2)* : tn+2(C) -»• to(C) given by / —> /|7n(VOJa:) ^s defined. We set r a (C) := im

Monodromy of Iterated Integrals

253

Observe that the diagram (kn+2y

Ipr t4(C)

(fc 4 ).

——^

Tofifoo (C)

commutes where the map pr\ is induced by the inclusion Va ^ P 1 (C)\{0,1, oo}. The map (£4)* is bijective because y4 = P 1 (C)\{0, l,oo}. Lemma 7.5 implies that the map pr is injective. The map (fcn+2)* is surjective by the construction. Hence both maps, (kn+2)* and pr\ are injective. Therefore we have proved the following result. Proposition 7.7. i) The 7r(yn+2)-conJugacy class of the monodromy homomorphism 0Xiyn+2 : 7Ti(3^n+2,x) —>• 7r(3^n_}_2) is determined by its restriction to7n(Va,x'). ii) The fK{ynjt2)-conjugacy class of the monodromy homomorphism Oxya : determined by the monodromy homomorphism ,l,oo},x')

§8. The Drinfeld-Ihara ^/5-cycle relation. In this section we show that the element which describes the monodromy of all iterated integrals on P1(C) \ {0,1,00} satisfies the Drinfeld-Ihara relation. 8.1. Introduction We recall that Yn = (Fl(Q)"

and yn = (P^QUO, l,oo})"~ ) " ~ 3 . Let

a,b,c e PX(C) be three different points and let (paib,c(z) = Y^L ' f ^ c * Th map ^ 4 j 5 : Y5 —>yy5 given i bby The $4,5(£l,

induces a bijection

The group £5 acts on Y5 by permutations. The action of £5 on I5 induces an action of £5 on ^5- The map a : ^5 —>• ^5, (r(s, t) = f | ^ , i j corresponds to the permutation a of I5 given by

254

Zdzislaw Wojtkowiak

Observe that the points -y/S-1

-\/5

are fixed by a. The one-forms ^ , ^ 3 , f, f , j-^ j - ^ p ^ f generate A\y5) and ^3, Let 50,^1, To, Xi and AT be their formal duals. There are two non-trivial relations in A2(y5): ds 5

dt t

dt t

— A—+ —A

ds — dt ds — dt ds —+ — A —, s—t s—t s

and

ds dt dt ds — dt ds — dt ds 5—1 t — 1 t — 1 S— t S—t 5— 1 Elementary computations of linear algebra imply that the subspace of H(y^)®2 is generated by A / \

~

—

_|_ -\~

—

A / \

•

I T"

[S^Tj + ^ i V ]

i = 0,l;

[Ti,Si] + [TuN]

<= 0,l;

[5 0 ,Ti]

A / \

;

and [SUTO]

where [A,B] = A® B - B ® A. We recall that P(y$) is a multiplicative group of the algebra of formal power series in non-commuting variables 5o, Si, TQ, T\ and N divided by the ideal generated by i?(3^s). The principal fibre bundle y 5 x p(y6) -+ y5 we equipped with the integrable connection given by the one form dt rds-dt

\

s

-t

dt\ _ dt\ —) t) tJ

Ar

, ds ^

5

a

ds

t

5

-

I

where T^ = —To — T\ — N. More simply, we shall write u instead of 8.2. Integration of u

Monodromy of Iterated Integrals

255

Recall that on P 1 (C)\{0,1, 00} we have 8.2.0

A ^ ( z ) • a^(Y,Z)

= AlSo(z)

(see Proposition 5.1).

Please notice that we are considering a^(Y, Z) as a power series in Y and Z. Thus we will be able to substitute Y and Z in a<^)(Y1 Z) by other pairs of letters. We shall denote the generators X, Y and Z from the beginning of section 6 by To + N, T\ and T^. The monodromy of Ao^1(z) is given by: (around 00) : Ao^1(z) —> Ao^1(z) • e~27rlTo°, (around 1) : A ^ ( z ) -> A ^ ( z ) • a ^ ^ , ^ ) • e " 2 ^ 1 • (a^fT^Too))- 1 (see Theorem 5.3). It follows from the definition of horizontal sections starting from a tangential base point, and also from the identity dz z

dz z— 1

dz z

dz dz z— 1 z

°°

that asymptotically at 00 and 1, we have respectively

8.2.1

A -,(«) ~ e ^ i.e. lira (A -, (z) • e ~ ^ i T ) T ~ ) = 1 z—>oo V ° ° 1 V

y

/

and

i.e. lim Z

z>l

Let Pe = (e, 1 + e) 6 ^5 where e is a small positive real number. Let Aps((s,t); path) be a horizontal section of u; such that Ape(Pe) = 1. Let 7 be a path in y5 from P£ to cr(Pe) = (e, 1/e) which is constant (= e) on the first coordinate. Assuming 5 = constant (= e) we have

Hence for small positive e, we asymptotically have

loo

e = 0

256

Zdzislaw Wojtkowiak

It follows from 8.2.0, 8.2.1 and 8.2.2 and Proposition 5.1 that for small positive £, we have 8.2.3

P

.W«);7)

o

Let p = <J4(7) •
which follows from Corollary 1.7 implies that

where L = Ape(cr(Pe);7). Let

It follows from 8.2.3 that £=0

The factors e< XT ( A ~ f ))M=<^))

ande

(/>*)^

canbeput

together in the product 0\J(L) •... • L because T^ = cr^(Ti) commutes with cr*(Ti) = 5o and a*(Too) = Si. After the calculations we get

f1+£

dt

dt\

f1/£ dt

Repeating the same argument for Si, Si + Ti + iV, T\ and So and passing to the limit e —> 0 we get

where a = a ^ ( T i , Too). The last formula can be written in the form a(S1 + TX + N,So) • aiSuTj

• a(Too,S1 + T1 + N)- a(SQ,Si) • a(TuT^

Monodromy of Iterated Integrals

257

since a*(S0) = T ^ , a* (Si) = Si + Ti + W, a*(r 0 ) = N, ^ ( T i ) = S o and Let ^5 : Ct - • 3^5 be given by ^5(21,22,23,24) = ^4,5(21,22,23,24,00). r-^2-)

Let (A;A -• be formal duals of ( —£ J

J

\

z

i

z

j

ip5*(Ai2) = So

. Then we have

) ij

-Si-To-Ti-N,

ip5*(Au) = So, ^5*{^23) = So + TO + N, ^5.(^24) = Si, ^5.(^34) = - 5 0 - S i - N. Using Vi : C* ->• ^5 given by il>i(z2,z3,z4,z5) get

= $4t5(oo, z2,z3,z4,

z5) we

^1*^23) = 5 0 + T o + iV,

<M>l34) = - S o - Si - JV, ^.(^35) =

- ^ - ^ - ^ ,

^1*^45) = ^ .

Set X^ := tpet(Aij) s = 1,5; then X15 = To. Hence finally we get a formula 8.2.4

a(Xi3,Xi4)-a(X24,X25)-a(X35,X13)-a(Xu,X24)-a(X25,X35) :

= 1.

If we use $2,4 C^ —> 3^5 given by $2,4(0? s, 1, ^, 00) = (5, ^) and repeat the calculations in y5 we get the same formula as before, but the X^'s names of 5o, S i , . . . are now different and the resulting formula is: 8.2.5 a(X 15 , X12) • a(X 23 , X34) • a(X 45 , Xi 5 ) • a(X i 2 , X23) • a(X 3 This is exactly the formula which appears in [12, p. 106]. Proposition 8.3. For any permutation a of five letters we have

ii)

258

Zdzislaw Wojtkowiak

where a(ij) = o~(i)o~(j). Proof. This follows from 8.2.4, 8.2.5 and Corollary 1.7.

0

Remark. The formulas of Proposition 8.3 take place in the group If we apply log we obtain formulas in the group 7r(y5). In the sequel we shall work in the group TT^S)We finish this section with a formula from which the Deligne ^/3-cycle relation can be obtained. The proof is an imitation of Deligne's proof. Proposition 8.4. Let a := log a. In the group 7r(3^) we have i) a(X23, X25)(-niX23)a(X35, X23)(-7riX35)a(X25, X35)(-7riX25) = -7riX14c and ii) a(X23, X2b)(niX23)a(X36i X23)(niX35)a(X25, X35)(7riX2b) = -7riX 14 . Proof. We prove only i) since ii) is similar. Let 5(x\,x2,x3,X4i,xr)) = (#i, £5, £ 2 , £4, £3). Then the induced map a : y$ -* 3^5 is given by a(s, t) =

( ¥ • ^T. ¥ )

and

°2M = fe I^)-

Let P

~ = ^ 1 - ') and P+ =

(r, 1 + r) where r is positive and small. Let Q_ = (—r, 1 — r) and Q+ = (—r, 1-hr). Let 7 be a path from P+ = (r, 1-hr) to (r, 1 + re i(v?+7r)) and let S" be a path [0, TT] 3 y? —> (—r, 1 -h re l ^ +7r ^). Let us consider the composition p = a(jf)oa(Sf)oa2(j)oa2(S)o1foS'oa(-f)oa{S)oa2(j/)oa2(Sf)o1oS. If we integrate the form u along this path and pass to the limit as r —> 0 we obtain the square of the left hand side of the equality in i). Let a be a loop in the opposite clockwise direction around (0,0) in the plane P = {(5, t ) e C 2 \ as + fit = 0}. The integration of the form u along a gives (-27rz)(So + iV + To) = (-27ri)X23. In the model of Y*/PGL2(C) in which the subspace {(xi,x2,x3,x±,x§) \ x\ = X4} of (P1 (C))5 degenerates to a point (for example for $2,5(0, s, 1, 00, t) = (5, £)), the path p is homotopic to a loop around one of the points (0,0), (1,1) or (00,00) in the plane passing through the corresponding point (0,0), (1,1) or (00,00) (the point (1,1) in the case of the model $2,5)- Hence the square of the left hand side 0 of the expression i) is also (—2ni) • X14. Corollary 8.5. For any permutation a of five letters 1,2,3,4,5 we have formulas i1) and ii1), which are obtained from formulas i) and ii) by replacing indices 1,2,3,4,5 by a (I), o~(2), 0£<7(i))*=i,...,5. The

Monodromy of Iterated Integrals

259

induced map a : J^ —> y$ satisfies (a* ® id)u = (id ® <J*)U>, which implies formulas i') and ii'). 0 Remark 1. We have X23 + X2*> + X35 = X14 in the Lie algebra Lie(^5 ). If we set X14 = 0 then the formulas i) and ii) reduce to the Deligne formula (see [D5]). Remark 2. Let f(z) = jz^- Let us return to the notation from section 6. We have f+(X) = Y and /*(T) = Z. Hence we get

(We consider the element a ^ as a power series in Y and Z.) Therefore we have Let p = [0,1] be a path from 01 to 10. The element a^(X,Y) is the result of integration from Ol to 10. The element considered by Ihara is the result of the Galois action on the path p composed with the path p~l. Both elements satisfy the Z/2, Z/3 and Z/5 cycle relations. The analogy is more striking when we consider torsors corresponding to the homomorphism %i : ^ ( P ^ C ) \ {0,l,oo},0l) -> ^(P^C) \ {0,1, oo}) and the associated groups (see Appendix A2).

§9. Subgroups of the groups of automorphisms. 9.0. We summarize here the notation which will be used in the rest of the paper. Let A; be a field of characteristic zero. We say that X is an algebraic variety defined over k if X is an algebraic scheme over Specfc. If A is a fc-algebra, we set XA := X x Spec A. We denote by X(A) the set of A-points of Specfc

X. Observe that X(A) = XA{A). We say that G is an affine algebraic (resp. pro-algebraic) group defined over fc, if G is an affine algebraic (resp. pro-algebraic) group scheme over Spec k. Let L be a nilpotent (resp. pro-nilpotent) Lie algebra over a field k of characteristic zero. We equip the Lie algebra L with a multiplication given by the Baker-Campbell-Hausdorff formula. We denote by n the obtained group. The group n is the group of fc-points in a connected affine algebraic unipotent (resp. pro-algebraic pro-unipotent) group scheme over A;, which we denote by II. Let Rbe &fc-algebra.We denote by II(i?) the group of itpoints of II. Let us set UR := U x SpecR. Observe that U(R) = UR(R). Speck

260

Zdzislaw Wojtkowiak

Let V be as in section 1.1. We shall assume that ^41(V) —> HpR(V) is an isomorphism. In 1.1 we defined the group TT(V). It is easy to see that TT(V) is the group of fc-points of a connected affine pro-unipotent pro-algebraic group scheme over k. We denote by II(V) the corresponding group scheme. (II(y) is Spec of the algebra of k-valued functions on n(V).) Let V = PQ \ {0,1, oo}. We recall that Lie(Vr) is the free Lie algebra over Q on X = ( ^ ) * and Y = ( ^ j ) * and that Z := -X - Y. We recall that L(V) is the completion of Lie(Vr) with respect to the filtration induced by the lower central series. Let us set V := [L(V),L(V)], L" := [L'.L1] and L2(V) := L(V)/L"We denote by ^ ( F ) the group scheme over SpecQ corresponding to L2(V). Let us set n2(V) = U2(V)(C). Notation.

Let w G L2{V).

If /(X, Y) = 1 4-

^2

a

We set wXlY^ n 7n

nmX Y

: = ( . . . (w,X).

..X)Y)

...Y).

is a power series, then we set

n=l,m=l

(wJ(X,Y)):=w+ n = l , 77i=l

9.0.1. Observe that (wXiY^)XaYb = wXi+aY^b in L2(V). This implies that the elements X, Y and (Y,X)Xi~1Yj-1 for i = 1,2,3,..., j = 1, 2 , 3 , . . . form a linear topological basis of L2(V). Let F2(Lie(V)) be the Lie ideal of L(V) topologically generated by Lie brackets in X and F , which contain X at least twice. Let us set L2(V) := L(V)/F2(L(V)). We denote by II 2 the corresponding group scheme over SpecQ. To simplify notation let us set TT2 :— II2 (C). If we replace X by Y in the above definition then the corresponding group of C-points we denote by w2. The elements X, Y and (YX)Yi (resp. (YX)Xi) i = 0,1,2,... form a topological linear basis of n2 (resp. m2). We denote by G m the multiplicative group scheme Spec(Q[t, t" 1 ]). 9.1.

Let us set

Aut*(7r2(V)) := {/ G Aut(7r2(y))|3 af G C*,/(X) = afX, f(Y) » a / y, f(Z) - afZ} (« is a conjugation by an element of (TT 2 (F), TT2(V)) and ~ is a conjugation by an element of TT2(V)). Let p : n2(V) -> ZA72 be the natural projection. The map p induces p* : Aut*(7r2(V)) -> Aut(ti72). Let C : G m (C) ->• Aut(tu2) be given by C*(X) = ^X, Ct(Y) = tF. We shall investigate liftings of C to Aut*(7T2(V)).

Monodromy of Iterated Integrals

261

Proposition 9.1. %) Let $ : G m (C) -» Aut*(7r2(l0) be given by

$t(X) = tX oo

Y

J^

n=2

*

where

c

i,j € C.

t+i=n * > 1, j > 1

T/aen $ 25 a homomorphism. ii) All homomorphisms $ o/G m (C) m£o Aut*(7r2(V)) s^cft thatp*o$ = C and $t(X) = £X are of this form. Hi) All one-dimensional subgroups G of Aut*(^(V)) which are projected onto C(G m (C)) by the map p* and preserve the subgroup of 7r2(V) generated by X are of this form. iv) The group G is defined over a subfield k of C if and only if all coefficients Cij are in k. Proof. The point i) is a straightforward verification if one uses 9.0.1. To N-l

show point ii) we can assume that &t(Y) = (tY,exp( ^ n=2

+

£

fijtyXiY*)).

]T} ci,j(l ~ i+j=n

Comparing coefficients of $ t

and * , ( * t ( y ) ) we get fid(t) 4- tNfid(s) = f^s) + ^ / , , , - W . This implies fij(t) = Cij (I — tN). Hence the point ii) is proved. If a subgroup of Aut*(?r2(V)) is one dimensional then the coefficients fij are algebraic functions of t. They cannot be multivalued functions because then the dimension of the subgroup would be greater than 1. Hence fij are Laurent polynomials of t. Now the point iii) follows from the proof of ii). The last point is obvious. <) In §13 we shall also need results about the subgroups of Aut(7r 2 / r n+i 7r 2) and Aut(7r 2 ). Let C : G m ( C ) -> Aut(7r 2 / r27r 2) be given by C[(X) = tX,C[{Y) = tY. Let (p n )* : Aut(7r 2 / r n +27r2 ) -> Aut(7r 2 / r27r 2) and p* : Aut(7r2) —>• Aut(7r 2 / r27r 2) be induced by projections of 7r2/rn+2n2 and ?r2 Onto 7T 2 / r 2 7 r 2.

Corollary 9.2. All one dimensional subgroups G of Aut(7r2/rn+27r2) (resp. Aut(?r2)) which the map (pn)* (resp. p*) projects onto C'(G m (C)) are of the form G = {ft eAut(7T 2 / r n +27r2 ) (resp.Aut(7r 2 ))

= tX,

n(resp.oo)

ft(Y) = tY+

J2

°i(t - ti+1)((YX)Yi~1)

with

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Zdzislaw Wojtkowiak

The group G is defined over a subfield kofCif

and only if all Ci 's are in k.

We shall denote the subgroup G of Aut(7T2/r»+27r2) (resp. Aut(?r 2 )) considered in the corollary by G(ci, C2,..., cn) (resp. G((ci)g 1 ) = G(ci, C2,..., cn,...)). Proposition 9.3. Suppose that for each n we have a point (c»j) i+j=n G Cn~1 and a sequence Z™,... , ZJ? of linear forms on Cn~1. Let Vn be a set of common zeros of Z™,..., ZJJ. Let us set L := {(Z™,..., ZJJ )n€N} o,nd c :=

i) Let G(c,L) := {/ G AnffaiV)) f(X) = tX, teC*, n=2

i+j=n

Then G(c,L) is a subgroup of Ant* ( Any subgroup G of Aut*(7T2(V)), which preserves the subgroup of 7T2(V) generated by X and whose projection into Aut(tu 2 ) is C(G m (C)) is of this form. Two subgroups G(c, L) and G(c', L1) coincide if and only if for each

ii)

Hi)

n ( * ; ) * £ £ +Vn = (dii)gi-h + VI The group G(c, L) is defined over the subfield k of Q if and only if for each n the affine space (cij) i+j=n 4- Vn is defined over k.

iv)

Proof. The point i) is a standard checking. Let G be as in the point ii). Let Gi := ker(p* : G —> Aut(o72)). Any element / € G\ is of the form f(X) = X, f(Y) = (y, exp( g n=2

£

f^X^)).

Observe that / ->

i+j=n t>i,i>i

N

((fij) i+j=n )n=2 3 N defines a homomorphism G\ -> J2 i

^

1

-^

1

' '

'

n=2

S

Ga(C) of

i+j=n

G\ into a direct sum of additive groups G a (C). Hence there are only linear relations between various / i / s . The group G is an extension of G m (C) by the pro-unipotent group Gi, hence there is a lifting C\ : G m (C) —> G of C. If we calculate the coefficients of C\t o f o Ci^1 we get that it can only be linear relations between the coefficients (/t,n-t)*=i,...,n-i- Hence for each n we have a finite number of linear forms Z",..., Z£ on C n - 1 such that

n—2

Monodromy of Iterated Integrals 0ij G C,V

V

n \
liWhn-l,p2,n-2,

263 • • •) = 0}.

Let L = {(/?,...,JJ n )nGi\r}- T n e group Ci(G m (C)) is as in Proposition 9.1.i) for some sequences (c™) z+j=n , n = 2 , 3 , , . . . Observe that G\ o i>

l,j>1

Ci(G m (C)) = G(c,L). Hence for dimensional reasons G — G(c,L). The points iii) and iv) are evident. <0 Let {ei)il1 be a sequence such that e» G {0,1} for each z. Let (c^)?^ be a sequence of complex numbers. For i = 1,2,...}, we set ^ i e ^ Z i ) = G ( C l , c 2 , . . . |e 1 ,e 2 , . . . ) : = { / = /t, ( e i A )£i € Aut(7r2)

= t •Y

with teC,fteC Replacing oo by n in the above definition gives a subgroup of Aut(7r2/rn+27r2), denoted by G ( c i , . . . , c n | e i , . . . , en). Let S((ei)f=1) be a number of €i equal to 1. Then dimG(ci,..., cn\eu ... ,en) = *((ei)? = i) + 1. Corollary 9.4. All subgroups of Aut(7r2/r«+27r2) (resp. Aut(7r2)), which the map (p n )* (resp. p*) projects onto C / (G m (C)) and which preserve the subgroup of ir2 generated by X are oftheform G ( c i , . . . , c n | e i , . . . ,e n ) (^res^. G((ci)g 1 |(ei)g : 1 )). The group is defined over a subfield k of C if and only if all numbers (1 — £i)ci are in k. We shall denote by Go the subgroup {ft € Aut(7r2/rn+27r2) ft(X) = tX, ft(Y) = tY,t e C*} of Aut(7r2/rn+27r2). map defined by 6(0) = 0,6(z) = 1 if z ± 0.

Let 6 : C -> {0,1} be a

Lemma 9.5. Le^ /o G Aut(7r2/rn+27r2) 6e siicft that fo(X) = aoX,fo(Y)

=

n

a0Y -f ^ ^ ( ( y X ) y * " 1 ) . Let G be the smallest closed algebraic subgroup i=l

of Aut(7r2/rn+27r2) such that Go C G and f0 G G. Then

Proof. Let Xt(^) = tX and xt(^) = tY and let fi = f0o (Xao)'1- Then = X and h(Y) = Y+J2 0SdQ1((YX)Yi-1). Let Gi C G be a subgroup consisting of h such that ft(X) = X, h(Y) = Y + fl

f3i((YX)Yi~l).

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Zdzislaw Wojtkowiak

Assume that for some i, f3f ^ 0. Then G\ ^ {Id} because f\ e G\. The subgroup G\ is isomorphic to a subgroup of ( G a ( C ) ) n , hence it is given by a finite number of linear forms. Let k — xt ° h ° (Xt)" 1 - Then k(X)

= X, Jfe(y) = y + f2tif3i((YX)Yi~1).

Let Z(a?i,...,x n ) be one of

2=1

linear forms defining G i . Then l(t(31,t2p2,. • .,tnpn) = 0 implies / = 0. Hence the /?i, /?2? • • • ,j8n» if they are non-zero, they are linearly independent. Therefore G = G ( 0 , 0 , . . . , o|<J(/3j),..., <*(#1)). <> Corollary 9.6. Le£ G C Aut(7r 2/r"+27r2) be the smallest closed algebraic such that the subgroups Go and G ( c i , . . . , c n ) subgroup of Aut(n2/^+2^2) are contained in G. Then G =

G(0,...,0\5(c1),5(c2),...,6(cn)).

Proof. One takes any element / of G(ci,..., cn) such that f(X) = aX and

§10. Torsors. Let G be a group. We say that a set T is a G-torsor if T is equipped with a free transitive action of G. We say that a subset 5 C T is a subtorsor of T if there is a subgroup H C G such that the natural action of H on S is free and transitive. Main example. Let Gi and G2 be two groups. Assume that G\ and G2 are isomorphic. Then the set of isomorphisms from G\ to G2, which we denote by Iso(Gi,G2), is an Aut(G2)-torsor. For any non-empty subset S C Iso(Gi, G2), the intersection of all subtorsors of Iso(Gi,G2), which contain 5, is a subtorsor of Iso(Gi,G2), which we denote by T(S). 10.1. Unipotent affine algebraic groups and torsors. L e m m a 10.1.1. Let G be an affine unipotent algebraic group over a field k of characteristic zero. Then there is an affine algebraic group Aut(G) over k such that for any k-algebra A we have

Let G\ and G2 be two affine unipotent algebraic groups over k. Then there is a smooth affine algebraic variety Iso(Gi,G2) over k, such that for any k-algebra A we have

Monodromy of Iterated Integrals

265

Proof. Let g be the Lie algebra of G. Let us equip g with the group law given by the Baker-Campbell-Hausdorff formula. The exponential map exp: g -> G is an isomorphism of affine algebraic groups. The group automorphisms of g coincide with the automorphisms of the Lie algebra g. One can easily give an ideal defining AutLie(s) in k[GL(g)]. One constructs Iso(Gi,G2) in a similar way. 0 We say that an affine algebraic variety T over A: is a G-torsor, if there is a morphism T x G —> T over Spec k, which defines a free transitive action o f G o n T (see [S] page 149). Let 5 be a closed subvariety of T and let if be a closed subgroup of G. We say that S is an i7-subtorsor or a subtorsor of T if S is an if-torsor under the natural action of H. Lemma 10.1.2. Let T be a G-torsor. Let T\ be an Hi-subtorsor of T and let T2 be an H2-subtorsor of T. Assume that T\ D T2 / 0. Then the intersection T\ D T2 is an H\ D H2-subtorsor of T. 10.1.3. Main example. Let G\ and G2 be two unipotent affine algebraic groups over k. Assume that there is an isomorphism (Gi)^ —> (^2)^ of algebraic groups. Then the algebraic variety Iso(Gi,G2) is an Aut(G2)torsor, if we equip Iso(Gi, G2) with the obvious action of Aut(C?2). Let k C C be a subfield of the field of complex numbers C. Let O : Gi(C) -> G2(C) be an isomorphism. Then 9 is a C-point of Iso(Gx, G 2 ). We denote by Z(Q) the A;-Zariski closure of © in Iso(Gi,G2) i.e. the smallest algebraic subset of Iso(Gi,G 2 ) defined over &, which contains 0 as a Cpoint. The unipotent affine algebraic group Gi is isomorphic as an algebraic variety over k to the affine space A™, hence 6 can be viewed as a C-point (9?;j)i
0

Definition-Proposition 10.1.4. LetT(Q) be the intersection of all subtorsors T defined over k o/Iso(Gi,G2), which contain © as a C-point. Then T(Q) is a G(0)-subtorsor oflso(GuG2) for some G(0) C Aut(G 2 ). Proof. The intersection of a family of algebraic varieties coincides with an intersection of a finite number of them. Hence it follows from Lemma 10.1.2

266

Zdzislaw Wojtkowiak

that T(Q) exists and it is unique. The group is also unique because it is an intersection of the corresponding subgroups of Aut(G2). 0 Lemma 10.1.5. Let G be a unipotent affine algebraic group over k. Then Aut(G) is an extension of an algebraic subgroup o/Aut(G ab ) by a unipotent affine algebraic group. Hence the group G(©) is an extension of an algebraic subgroup of Aut(G ab) by a unipotent affine algebraic group. Proof. Let g be the Lie algebra of G and let (Pg); be the filtration of g by the lower central series. Any automorphism of the Lie algebra g preserves the filtration and the induced automorphism of I y / r i + 1 g is determined by the automorphism of g ab = F Bfr2g- Hence AutLie(#) is an extension of a closed subgroup of GL(g3h) by a unipotent group. The lemma follows from the identification of Aut(G) with AutLie(s) by the exponential map exp : g -^ G. 0 Lemma 10.1.6. Assume that G(0) is an extension ofGm (orG such that H1(Gsl(k/k),G)= 0) by a unipotent affine algebraic group N. Then T(Q) has a k-point. Proof. It follows from [S] Proposition 4.1 that ^(Gal^/fc),TV) = 0. It follows from [S] Proposition 2.2 and the assumption of the lemma that = 0. Prop. 1.1 of [S] implies that T(S)(k) / 0. 0 Let P(G) be a filtration of a group G by the lower central series. Let us set G (i) := G/ri+1G- The isomorphism 6 : Gi(C) -> G2(C) induces isomorphisms 9 ( i ) : G^(C) -> G^(C). Let k < i. The projections Gf -> G{k) for j = 1,2 induce pi:Iso(G^\G(^)-^lso(G[k\G{2k))

and p(2)l : Aut(G^) -> Aut(G^).

Lemma 10.1.7. We have

a) pUne^)) = r(e(fe)), Hi)

W

()

Proof. In the point i) ( ) means thefc-Zariskiclosure and we omit its proof because we do not need this fact later. Let us set p = p\ and p' = p(2)lk* Observe that the image pf(G(@^)) of the group G(@^) by the morphism p' is a closed subgroup of Aut(G;T ) defined over &. This implies that p(T(0W)) is a closed subvariety of lso(G[k\G(2k)) and a p'(G(e^))-torsor defined over k. This torsor contains 9 ^ as a C-point, so we have and

Monodromy of Iterated Integrals

267

Let P (resp. P1) be the projection p (resp. p') restricted to T ( 0 ^ ) (resp. G(0«)). Then P" 1 ^©***)) is a P / - 1 (G(9^)))-torsor defined over fc, which contain 0 « as a C-point. Hence we get p - 1 ( T ( e ^ ) ) = T(©W) and P ' - ^ G i e ' * ' ) ) = G(0W) This implies that p(T(0«)) = T(e<*>) and

10.2. AfRne pro-algebraic pro-unipotent groups and torsors. 10.2.1 We assume that G = limG^, where the groups G ^ are affine iinipotent algebraic groups over k and the morphisms GW —> G^) are also over k. We assume further that G^ = G/YiJtlG- Finally we assume that the Lie algebra g of G is finitely presented, i.e. that for i big enough the number of relations defining O/r*"1"1^, of degree less than i + 1 does not depend on i. 10.2.2. The condition that G is finitely presented implies that for i big enough the morphisms Aut(G^+1^) —>• Aut(G^) are surjective. We set Aut(G) :=limAut(G (i) ). Similarly, if Gi and G2 satisfy 10.2.1 and if there is an isomorphism (Gi)*. —> {G2)k of affine pro-algebraic pro-unipotent groups, then the morphisms Iso(Gi~ ,G2 ) —> Iso(Gi ,G^ ) a r e surjective for i big enough. We set Iso(Gi,G2) := limIso(Gi JGJ ). i

Observe that Iso(Gi,G2) is an Aut(G2)-torsor defined over k. 10.2.3. Examples of groups satisfying 10.2.1. 1) Let F b e a smooth algebraic variety over k. Let us set

(see §2 and [Wl]). Then G satisfies 10.2.1. 2) Let V be as in 1.1. Then the group scheme H(V) satisfies 10.2.1. Lemma 10.2.4. Let G be as in 10.2.1. The affine group scheme Aut(G) is an extension of a closed subgroup of GL(Gah) by an affine pro-unipotent pro-algebraic group. This lemma follows from Lemma 10.1.5 and the definition of Aut(G).

268

Zdzislaw Wojtkowiak

10.2.5 Let Gi and G2 satisfy 10.2.1. Let O : GX(C) -> G2(C) be an isomorphism. Then 6 = I i m 9 ( i \ where 9 ( i ) : G^(C) -> G^(C) are i

isomorphisms for all i. Let us set Z(@) := lnnZ(6 ( i ) ), T(0) := limT(9 (i )), G(9) := UrnG(6 w ). Then T(9) is a G(9)-torsor. §11. Torsors associated to non-abelian unipotent periods. Let X be a smooth quasi-projective algebraic variety defined over a number field k. Assume that X has a fc-point x. Let us fix an embedding k c-> C. In [Wl] §7 (see also 2.0 in this paper) we have constructed affine pro-algebraic pro-unipotent finitely presented group schemes 7rf (X(C),x) and ?rPR(X, x) over SpecQ and Spec k respectively. We set

Then we have:

irf(X(Q,x) =lim7r?(X(C),z)(n) and n? In [Wl] Proposition 7.5 we have also constructed a homomorphism (called Boain[Wl])

such that the induced map on C-points,
-+ 7T»R(X,

is an isomorphism. We have cpx = lim<^", where y?" : 7r^(X(C),a;)(n^(C) —>• n

x)(n\C) is induced by (px. For each n we have an Aut(7rPR(X, x)^)torsor Ison := Iso(7rf (X(C),x)W x Jb, 7rf^(X,x)( n )). Applying the construction from §10 to the isomorphism x) is a subgroup of Aut(?rPR(X, x)^). We set and TTP R (X,

Monodromy of Iterated Integrals

269

We have projections pn : Aut(7r? R (A»< n >) Aut(7rPR(X, x)(n) and p : Aut(7rP R(X,x)) -> Out(7rP R (X,z)). Definition 11.1. i) ii)

is the image of G((p%) in

The group Q{X,x)W

The group Q(X,x) is the image of G((px) in Out(?rPR(X, x)).

Let x and y be two fc-points of X. Let 7 be a path in X(C) from x to y. Then 7 induces an isomorphism c 7 : TTP R (XC,X) —» 7rPR(Xc,y). The induced isomorphism of outer automorphisms groups (c 7 ), : Out(7rP R (X c ,z)) -> Out(7rPR(Xc,
The isomorphism is unique up to conjugation by elements of Hence the induced isomorphism (c,), : Out(7T?R(X,z)) ->

nfR(X,y).

Out{nfR(X,y))

is canonical, in other words it does not depend on the choice of a fc-point of Spec ( ^ ( p - " 1 ^ ,
irf(X(C),x)(C)

^

7r?R

where c 7 and c7 are induced by the path 7. Observe that c7 = cv oconj(#), where 77 is a fc-point of Spec (H^R((X; x, y))j and conj(#) is a conjugation by an element g € 7rfR(X,x)(C).

This implies the following result.

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Zdzislaw Wojtkowiak

Proposition 11.2. The groups Q(X,x)^ and Q(X,y)(n^ coincide under the canonical isomorphism (c^)*. The groups Q(X,x) and Q(X,y) coincide under the canonical isomorphism (c^)*. We shall denote these groups by and GDR(X) respectively.

§12. Torsors associated to the canonical unipotent connection with logarithmic singularities. 12.0. Let V be as in section 1.1. We shall assume that A1^) -» H]^R(V) is an isomorphism. We assume that V has a fc-point x. We recall from Corollary 2.4 that we have an isomorphism u : Trf^V, x) —> U(V) (U(V) := Spec(Alg7r(V))). Let u(C) :
and Let ^ a;^ 0 •. . 0 ^ f c be a cocycle representing a class c in if£ R (/9 # ~ 1(x, #)). (We recall from §2 that n?R(V,x) := Spec (flS R (p— ^ ^ x ) ) ) . ) We can The assume that all Uit are one-forms (see 2.2.1). Let a e TTI(V(C),X). value of c on bDR(a) is given by the following formula:

12.1.

c(bDR(a)):=Y
-->«>ik (see2.8).

The homomorphisms 6Q and 6^ are defined in the same way. We want to calculate G(<^x)-torsor T((px) associated to the comparison homomorphism

Lemma 12.2. Tfte G(ipx)-torsor T(tpx) is isomorphic to G(u(C) o ipx)torsor T(u(C) o ipx). Proof. It follows from the fact that u :

*.

TT^R(V^X)

-> Ii{V) is defined over

0

We shall relate the G(u(C) o (px)-toTSOT T(u(C) o cpx) to the monodromy homomorphism 9xy : 7Ti(V(C),:r) -> II(V)(C) of the form UJV from §1. Lemma 12.3. We have

Monodromy of Iterated Integrals

271

Proof. The isomorphism u is induced by Uix ®.. .<8>u)ik —> (X^ ®.. .(&Xik)* (see corollary 2.4.). The value of the class c on bDR(a) is given by 12.1. The value of £(* i f c , . . . , c ^ (see Definition 1.3 and Proposition 1.8). It follows from [Chi] (1.6.2) that It follows from the definition of the homomorphisms bDR and b^ that

Let R be a fc-algebra. Let iR : Trf (V(Q,ar)(Q) -> Trf ( ^ ( C ) , ^ ) ^ ) be the inclusion of Q-points into ii-points. We have ic ° bq = b^. We set We define a functor Xy on fc-algebras in the following way: IV(R) := {/ : TTI(F(C),X) -> II(Vr)(/2) | 3 an isomorphism / : **(V(Q,x)(R)

-> U(V)(R), fobBR = / } .

Observe that / is uniquely determined by /. The functor Xy is represented by an affine pro-algebraic scheme overfc,which we also denote by Xy. Moreover Xy is an AutII(V)-torsor. Observe that 6xy : TTI(V(C),Z) -> n(V)(C) is a C-point of Xy because 6xy = (u(C) o ipx) o b^. We denote by T{0xy) the smallest subtorsor defined over k of Xy such that 6xy G T(6xy)(C) (i.e. the intersection of all subtorsors of Xy defined over fc, which contain 6xy as a C-point). The corresponding group we denote by G(9xy). Lemma 12.4. The G(9xy)-torsor T(6xy) ipx)-torsor T(u(C) o ipx).

is isomorphic to the G(u(C) o

Proof. First we notice that the Aut(II(V)-torsor Xy is isomorphic to the Aut(II(Vr) torsor Iso(?rf (V(C),a;) x k,H(V)). The isomorphism is given by / -> / . The equality 8xy = (u(C) o ipx) o bB implies the lemma. 0 Corollary 12.5. Let V be a projective line over k minus a finite number of k-points. Let S be a loop on V(C) around a missing point. Then the element bDR(S) e K?R{V,V){C) is conjugated to s~2ni, where s e 7r?R(V,v)(k). Proof. It follows from Lemma 12.3 and Corollary 5.4.

0

12.6. Let V be a projective line P£ minus a finite number of fc-points. If x is a tangential base fc-point v then the groups ?rf and ?rfH are not defined in [Wl]. We set ir?(V(C),v) := (iri{V(Q>v)h -Malcev rational completion of 7ri(V(C), v). For the Q-algebra R, the map 6f : TTI(V(C), V) -> n?(V(C),v)(R) is the natural map ni(V(C),v) -> (ni(V(C),v))Q(R) (see

272

Zdzislaw Wojtkowiak

§A.l). The monodromy homomorphism 0$y • fti{V(C),v) —» Il(V)(C) is the Malcev C-completion (see §A.l), hence there is an isomorphism (p$ : ?rf (F(C), v){C) -> 11(7) (C) such that (fyob£ = 0€y. The G(0ity)-toT8OT T(9$y) is defined as above. We set also ?rfR(V, v) := H(V). The isomorphism u is the identity and the G(^y)-torsor T(6^y) is isomorphic to G(^)-torsor T{{p$). §13. Partial information about GDR(PQ \ {0, l,oo}). 13.1. Let V = PQ \ {0,1, oo} and let v be a Q-point of V or a tangential base Q-point of V. For any Q-algebra R we set: T(R):={(x,y,z)€(H(V)(R))3\ 3a eR*,

x = aX,

y « aY, z ~ aZ, x-y-z

= 0}

and

Auf (n(V))(ii):= aY,

f(Z)~aZ}.

The functors T and Aut*(II(V)) are represented by affine pro-algebraic schemes over Q, which we also denote by T and Aut*(II(Vr)). Aut*(II(Vr)) is an affine pro-algebraic group scheme, a subgroup of Aut(II(V)). Moreover T is an Aut*(II(Vr))-torsor. (Let (x, y, z) and (xf, y', zf) be in T(T^). The map x —> x', y —> y' can be extended to an automorphism of the Lie algebra, which is also an automorphism of the group U(V)(R). Hence the action is transitive.) If we replace the group U(V) in the above definitions by n 2 (F) (resp. II 2 ), then we obtain an Aut*(n2(Vr))-torsor T2 (resp. Aut*(II2)-torsor T 2 ). Let us fix generators 5Q, 5^ and S^ of TTI(V(C), V) which are loops around 0,1 and oo respectively, such that Sf0 • S[ • S'^ = 1. Lemma 13.1.1. The Aut* (U(V))-torsorT is a subtorsor of the Aut(U(V))torsor Xy. Proof. Let (x,y,z) G T(R). Let / : ni(V(C),v) -> U(V)(R) be given by f(S'Q) = x, f(S[) = y. The homomorphism 6f : KI(V(C),V) -> Trf (F(C), v)(R) is the Malcev ^-completion of TTI(V(C), V) (see §A.l), hence there is an isomorphism / : 7rf (V(C), v)(i?) -^ II(Vr)(i2) such that /ofeg = /. Therefore f £lv(R). 0 Let 0V : 7Ti(V(C),v) -> n(V)(C) be the monodromy homomorphism of the one form uv- Observe that the triple (0v(SfQ), 0v(S[), ^ ( S ^ ) ) £ T(C). Let Tf(9v) be the smallest subtorsor defined over Q of T such that the triple

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(01/OSo), ev(S[), OviS'n)) G T(C) is a C-point of T'(0V). Let G'(0V) be the corresponding group. We recall that in §12 we defined G(0v)-torsor T(6v). Lemma 13.1.2. The G(0v)-torsorT(0v) and the G'(0v)-torsorT'(0v) are equal if we identify T with a subtorsor of Xy via the map from Lemma 13.1.1. We shall use the same notation T(0V) and G(0V) for both torsors and both groups. 13.2. We recall that the monodromy homomorphism 0_A> : 7Ti(V(C), ~ot) —> Tl(V)(C) is given by 0^(SO) = (-2ni)X 1

• (-2ni)Y . <(X, y) 1

aJ?(X, Z)

where aJ?(X,F) 6 (n(V),ir{V)) and 5 0 ,5i,5oo are as in the figure in 0.2. The triple (0^(S o ),0_£(Si),0^(Soo)) ^ T(C). Hence we get the G(0^)torsor T(0__A.), a subtorsor of T. Proof of Theorem B from the introduction. Let u be a Q-point of V. It follows from Lemmas 12.2 and 12.4 that the group schemes G(tpv) and G(0V) are isomorphic. The isomorphism is induced by the isomorphism u : TTP^V, V) -¥ n(V). Hence the images of the groups G((pv) and G(0V) in the groups of outer automorphisms are also isomorphic. The homomorphisms 0V and 0_A, are conjugate; hence the images of groups G(0V) and G(0_^) in Out(II(Vr)) are equal. Observe that the map Aut*(II(Vr)) ->• O\it(U(V)) is injective. Hence fon(V) and G(0—±) are isomorphic. 0 13.3. Let 0 : 7n(V(C), ot) -> n 2 (V)(Q (resp.02 : TTI(V(Q, ot) ^ n 2 (C)) be the composition of 0_^ with the projection on n2(Vr)(C) (resp.n2(C)). The triple (0(5o),0(5i)1°0(5oo)) e T2(C) (resp. (02(5O),02(51),02(5OO)) € T 2 (C)). Repeating the construction from 13.1 we get a G(0)-torsor T(0) (resp. G(02)-torsor T(0 2 )), a subtorsor of Aut*(n 2 (^))-torsor T2 (resp. Aut*(n2(C))-torsor T 2 ). The projections p : U(V) -> n 2 (F) and p 2 : II(Vr) -^ n 2 induce homomorphisms of group schemes and (p2)* : Aut*(n(F))-> Aut*(n 2 ) and morphisms of torsors compatible with homomorphisms of group schemes P*:T^T2

and (p2)* : T-> T' 2 .

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Lemma 13.3.1. We have

and

>)) = G(e2). This lemma follows from Lemma 10.1.7. Observe that the group G(9) is the image of the group G(Q—±). This implies Theorem C from the introduction. Corollary 13.3.2.

The group G(62) is isomorphic to a quotient of

13.4. Below we shall calculate the G(02)-torsor T(0 2 ). We have II2 = lim (n 2 /T n + 2 II 2 ). If in definitions of T and Aut*(II(V)) in 13.1 we replace the group U(V) by n 2 /r n + 2 II 2 , then we get an Aut*(n 2 /r n + 2 n 2 )-torsor T2. Let 6 n be the composition of 02 with the projection onto II 2 /r n + 2 II 2 (C) = n 2 (C)/r"+ 2 II 2 (C). The triple (e n (5 0 ),e n (5i),e n (Soo)) € 7^(C). By definition Tn := T(Qn) is the smallest subtorsor of T2 defined over Q which contains the triple as a C-point and Gn = G(On ) is the corresponding group, a subgroup defined over Q of A u t * ( n 2 / r n + 2 n 2 ) . Let p^ T2+ x and p^1 : A u t *(( n) 2 / r n + 1 + 2 n 2 ) -^ Aut*(n 2 /r n + 2 n(2 )) be induced by the projection n 2 / r n + 1 + 2 I I 2 -> n 2 / r n + 2 n 2 . It follows from Lemma 10.1.7 that

13.4.1.

PZ+1(Tn+1) = Tn

and P£ + 1 (G n + 1 ) = G n ,

We recall that by definition T(02) := limT n and G{62) := limG n . n

n

We recall that the group of Q-points (II 2 /r n+2 II 2 )(<2) is a Lie algebra over equipped with the multiplication given by the Baker-Campbell-HausdorfT formula. A linear basis of this Lie algebra is given by X, Y, (Y,X),..., (Y,X)Yn~1. The homomorphism 62 : 7Ti(F(C), ot) -> II2(C) is given by 92(S0) = {-2m)X and 0{Si) = {-2m)Y + § {2-Ki)C>{k){{YX)Yk-1) (see §6). k=2

0. Calculations of To and Go-

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We have 60(50) = (-2ni)X and 60(51) = (-2ni)Y. Observe that {2m) is not afc-throot of a rational number for any k = 1,2,3... . This implies that T0(C) - {aX,aY | a G C*} and G0(C) - {fa I fa(X) = aX, fa(Y) = aY I a G C*} C Aut(n 2 (C)/r 2 n 2 (C)). 1. Calculations of Xi and Gi. We have 6i(5 0 ) = {-2m')X and 0i(Si) = (-27ri)y. Hence we get aeC*} and Gl(C) = {/a I / a W = <*X, /a 0 0 = ^

I * G C*}.

2. Calculations of T2 and G 2 .

We have 0 2 (5 O ) = (-2ni)X, 6 2 (5i) = (-2ni)Y + (2ni)C(2)((YX)Y). Observe that (2TTZ)C(2) = ^(-2TTZ) 3 . This implies that T2(C) = { a l , a 7 + ^ a 3 ( ( y i ) y ) | a G C* } and G2(C) = {/a I fa(X) = OLX, fa(Y) = aY | a e C*}. 3. Calculations of T3 and G3. We have 6 3 (5 0 ) = (-2TTZ)X, 6 3 ( 5 I ) = (-27ri)y + (27ri)((2)((YX)Y) + +(27rz)C(3)((rX)F2). Assume that dimG 3 = dimT3 = 1. Then it follows from 13.4.1 and Corollary 9.2 that G3 = G(0,0,c3) where c3 e Q. As T3 is a one-dimensional variety we have T3(C) = {aX, aY + -^a3((YX)Y) + P3((YX)Y2) I a e C*, fteC, p(a,/?3) = 0} for some Laurent polynomial p(x,y) € Q[x, ^,2/]- Observe that to each value of a corresponds exactly one value /33(a), because T3 is a G3-torsor. Hence /33(a) = p(a) where p(x) 6 Q[x, i ] . If (aX1aY+^a3((YX)Y)^p(a)((YX)Y2) G T3(C), then ((aOX,(aOy+^(aO 3 ((yX)y)+(p(a)^ 4 + C 3 ^-^ 4 )a)((yX)y 2 )) G T3(C). Hence p(at) = p(a)t4 + c3(t — tA)a. Therefore p(x) = ax 4 + c3x. The equality (27rz)C(3) = a(—2TT2')4 + C3(-2TT2') implies a = 0 and c3 = — C(3). Therefore we get dimG 3 = 1 if and only if £(3) G Q. Therefore if C(3) ^ Q (one knows that £(3) is irrational) then dimG3 = 2, T3(C) = { a X , a F + ±a3((YX)Y)

+/? 3 ((FX)F 2 ) | a G C*, /% G C}

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and G3 = (0,0,0 | 0,0,1). n. Calculations of Tn and Gn. We have 9 n (S 0 ) = {-2ni)X, n-l

k=2

Assume that Gn-\

= G(0,0, C3,0, C5,0, C7,... | 0,0,63,0,£5,0,67,...)

(see Corollary 9.4), where c\ = c2 = c 4 = . . . = c2k = . . . = 0, C2fc+i =

-C(2fc 4-1) and e2k+1 = 0 if C(2fc + 1) G Q, and c2fc+i = 0 and e2k+1 = 1 if -1) g Q. Then the torsor T n _i corresponding to the group G n _i is given by n-l

n-l

((1 - £fcHa + ek/3k) ((YX)Yk-1)

| a € C , VA; ft €

Assume that n = 2p. Then C(n)(27ri) = r n (-27ri) n + 1 , rn e Q. This equality, the fact that (2ni)n £ Q and the property 13.4.1 imply that Gn = G(0, 0, C3, 0, C5, . . . , Cn-i, 0 I 0, 0,ff3,0, . . . , £„_!, 0) and

+ ekfa) ((yX)y fc - 1 ) | a 6 C*, V ft G c } . fc=3,fc-odd

fe

Assume that n = 2p+ 1. Assume that dimG n = dimG n _i. Then G n - G(0,0, c 3 ,..., 0, c n | 0,0, e 3 ,0, e5 , • • •, 0,0)

(by Corollary 9.4)

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and n-l

Tn(C) = {aX, aY +

£

rkak+1((YX)Yk-1)

+

k=2,k-even n-l

((l-ek)cka + ekpk)((YX)Yk-1)+pn((YX)Yn-1)

]T

with

k=3,k-odd CKGC*;

Vfc,fteC; p{a,ez-faeb-fa

.. .,&) = 0}

for some polynomial p(x, y 3 ,2/5,..., yn) € Q[x, ±, 2/3, y 5 ,..., yn). Assume that for some a$ there are two different j3n. Then there is g in Gn(C) such that g(X) = X, g(Y) = Y + . . . + bn((YX)Yn-1) with bn ^ 0. Then it follows from the proof of Lemma 9.5 that Gn — G(0, 0, C3,0,..., 0, 0 | 0,0, £3,0,..., 0,1) and dimG n = dimG n _i + 1 . Hence for any a G C* there is a unique /3n corresponding to that a. Observe that /?„ is an algebraic function of a, /?3, /?5,... , which depends only on a. Hence /3n = p(a) for some p(x) G Q[rr, ^] • Choose any / G G n (C). Then f(X) = tX and n-l

f(Y) = tY+ Yl

((! " ^)C*C "

for some t G C*, 63,65,..., 6 n _ 2 G C. Acting by / on any element of Tn(C), we get n-l

{{crt)X, (at)Y+

Yl

rk(at)k^1((YX)Yk-1)

fc=2, fc-even

n-l

+

J2

(«(1 - ek)ck(t - t^1) + aekbk + (1

k

k=3,k-odd

+ ek(3ktk+1)((YX)Yk-1) + ( a C n ^ - r + ^ + ^ Therefore we get p(at) = cna(t - tn+1) + p(a)tn+1. This implies p(x) = ax n + 1 + c n x. The equality C(n)(-27ri) = a(2ni)n+1 cn(27ri) implies p(x) = cnx and cn = — C(^). Therefore we get dimG n = dimG n _i

if and only if C(n) G Q.

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The final result is the following. Proposition 13.5. Let G = limG n . Then G(62) = G and G = n

G(0,0, c 3 ,0, c 5 ,0, c 7 , . . . , 0, c2jfe+i, 0 . . . | 0,0, e 3 ,0, e 5 , 0 . . . , 0, e2fc+i, 0 . . . ) , :

£2fc+i = 0 if and only if ((2k + 1) G Q, and then c2fc+i = — ((2k + 1). The torsorT(62)(C) is given by oo

{aX,aY+

J2

oo

k+l

k 1

rka ((YX)Y - )+

^

((

'M I a € C , for all Jk, /3fc e

where ((2k) = -r2k(27ri)2k, andc2k+i = —C(2*:H-1), ^2fc+i = 0 i/C(2fc+l) G = 0, Corollary 13.6. T/ie ^ro^p G contains the group H := {/a | fa(X) = aX, fa(Y) = ctY \ a e C*} if and only if all numbers (,(2k + 1) are irrational. Proof. It follows from Proposition 13.2 that all £(2fc + l) are irrational if and only if G = G(0,0,0,...,0,... | 0,0,1,0,1,0,.. .0,1,0,1,0,...) (all a = 0, e\ = 0, all 62k = 0 and all £2fc+i = 1 for fe = 1,...). Then we have H C G. 0,...). Then there exists f € G such that /(X) = a l and

where all X2A;+I / 0, Corollary 9.5 implies that G = G ( 0 , 0 . . . , 0 , . . . | 0 , 0 , 1 , 0 , 1 , . . . 0 , 1 , 0 , . . . ) (all a = 0, ex = e2fc = 0, e2fc+i = 1 for fe = 1,2,...). 0

Corollary 13.7. Le£ ^ 6e the smallest closed subgroup of Aut(?r2) defined over Q, containing G and H. Then Q = G(0,0..., 0,... | 0,0,1,0,1,0 ...) fa// Ci = 0 , e i = e2fc = 0, e2k+i

= 1 for k =

1,2,...).

Proof. The group Q contains / such that f(X) = X and

k=l

where all x2k+i ^ 0. Lemma 9.5 implies the corollary.

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279

13.8. Proof of Theorem D in the introduction. We recall from §6 that the monodromy homomorphism

is given by S o - • (-27rt)X, i=O,j=O

(see §6, formula (7)). It is an observation of Drinfeld that the numbers satisfy the equation

-1+

( n>0,m>l

k=2

(see [Dr]). Therefore we can describe the monodromy homomorphism 0 in the following way

k=2 ?+j=k

J

Let us temporarily write X" for the torsor given in Theorem D i) and Gf for the corresponding group. Then T(6) is a subtorsor of T' and the group G{6) is a closed subgroup of G' because (0(So),0(Si)) G T'(C). The image of the group G' in Aut(n(F)/r n+2 II(V r )) has the same dimension as the subgroup Gn of A u t ( l l 2 / r n + 2 n 2 ) . The group G(6) projects onto G n , hence Gf = G{0) andT' = T(0). The point iii) of Theorem D and Corollary E follow from Corollary 13.7 and Corollary 13.6 respectively. §14. The group G*DR(V) for pointed projective lines and for configuration spaces. In this section we shall translate the results about the monodromy representations from §§5-8 into results about the group GDR(V). Let V = P£\{ai,..., a n +i} and let v be a Appoint of V. We can assume that a\ = 0, a2 = 1 and a n +i = oo. We recall that Lie(V) is a free Lie algebra over k on Xi := (z^_za. )*, 2 = 1,2,..., n. Let us set Xn+\ = — ^^=1 Xi. Let R be a fc-algebra. Let us set

Aut* (U(V))(R) := {/ G Aut(U(V)R) | 3a/ G i?*, /(*i) = afXu f(X2) « a/X 2 , /(Jffc) ~ a/^ib * = 3, ...n + l}.

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Here « means a conjugation by an element of U(V)(R) with the zero coefficients at X\ and X2 and ~ means conjugation by an element of U(V)(R). Similarly for yn = (P x (C)\{0, l,oo})"~ 3 we set Aut*(U(yn))(R)

:= {/ G Aut(n(y n )R)| 3a/ € JT, /(X i f i ) - a , * ^ - } .

(The elements Xij are as in §7.) These functors are representable by affine pro-algebraic group schemes over k and over Q, which we denote by Aut* (11(10) and Aut*(lI(Xi)) respectively. Let us set Out*(11(10) := image(Aut*(n(Vr)) -> Aut(n(F))/Inn(n(F))) and

Out*(n(yn)) := Aut*(n(yn))/inn(n(yn)). We shall define an Aut* (11(10 )-torsor T and the Aut*(n(^ n ))-torsor Tn as follows. Let S i , . . . , S n + i be geometric generators of TTI(V(C),V), i.e. loops around a i , . . . , an_|_i respectively. For any fc-algebra R we set T(R) := {/ €Hom(in(V{C),v);Il(V)(R)) \ 3af G IT, f(S1) = f(S2) w afX2, f(Sk) ~ a/Xfc * = 3,..., n + 1} and T"(i2) = {(f eHom(vi(yn{C),y);Il{yn)(R))

\ 3a,, £ R* such that

(The elements A^- are as in §7.) Proposition 14.2. We have GDR(V)

C Out*(R(V)).

Proof. Let us choose the vector 0i as a tangential base point. Let S\ be a loop around 0 as in the picture of §4. Let us choose a tangent vector Vk at each point a&. Assume that V2 = T^. Let F be a family of paths from the base point to V2, -. •, vn+i such that 72 is the interval [0,1]. Let Si, S2,..., S n +i be a sequence of geometric generators associated to F. Let 0_^ : TTI(V(C), "ot) —f II(y)(C) be the monodromy homomorphism. It follows from Corollary 5.4 and Theorem 6.1 that G T(C).

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281

Hence G(0^) C Aut*(n(F)). This implies that QDR{V) C Out*(n(F)) if we identify n^R(V,v) and II(F) by the homomorphism u from Corollary 2.4. 0 Observe that Tn is a Aut*(n(3^n))-torsor. Hence T n /Inn(n(^ n )) is an Out*(n(^ n ))-torsor. Proposition 14.3. n)c

Out* (n(yn)).

Proof. Corollary 7.3 implies that 0y^yn G T n (C). Hence the proposition follows. Proposition 14.4. For n > 4 we have isomorphisms

induced by the projection yn+\ —> yn.

Proof. The proposition follows from Proposition 7.4, Lemma 7.5 and Corollary 7.6. 0 Observe that 34 = PQ \ {0,1, oo}. Hence we have the following corollary. Corollary 14.5. For n > 4 we have GDR(yn) « GDR{FQ \ {0,1, oo}). The group H(V) is a normal subgroup of n(3^ n+2 ). Hence n(>?n+ 2) C Aut(n(F)). Proposition 7.7 implies the following result. Proposition 14.6. We have GDR(V)

- n(yn+2)/n(yn+2)«

gDR{v\i \ {o, 1, oo}).

§15. Conjectures. Let X be a smooth quasi-projective algebraic variety defined over a number field k and let a; be a fc-point or a tangential base Appoint. Denote by 7rf*(X xfc,x)i the pro-1 completion of the etale fundamental group o f X x L k

k

Let &x ' Gal(k/k) —> Out(7rf*(X x k,x)i) be the natural representation of k

the Galois group into the outer automorphisms of the pro-1 completion of the etale fundamental group.

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The monodromy representations of iterated integrals, i.e. the homomorphisms TT? R (X,X)(C) have many properties analogous to the action of Gal(k/k) on etale fundamental groups. We give some examples. Example 1. Let X be the projective line minus a finite number of kpoints. Let 5 be a loop on X around a missing point. If a G Gal(k/k) then a(S) is conjugate to Sx^a\ where x is the cyclotomic character (see [13]). It follows from Corollary 12.5 that (px(S) is conjugate to an element s27™, where «€7r? H (-Y,*)(*). Example 2. Assume that X is a smooth projective variety and D is a divisor with normal crossings in X. Assume that X := X \ D is defined over a field k. Let S be a loop in X around an irreducible component of D. As before a(S) ~ Sx^ in the case of a Galois action. It follows from Corollary 7.2 that ipx(S) ~ s2nt, where s £ /K^R(X,x)(h) and X is a configuration space of n points in C (see §7). Example 3. The element aJ?(X, Y) satisfies Z/2, Z/3 and Z/5 cycle relations. The element considered by Ihara in [12] satisfies analogous relations, (see also Remark 2 at the end of §8.) Example 4. Chudnowsky has shown that the transcendence degree of the field generated by abelian periods is 2 if X is an elliptic curve with complex multiplication. The existence of a non-trivial endomorphism of X implies that the image of Gal(fc/fc) is contained in a two dimensional subgroup of

GL(Hlt(X x *),). In the second part of this paper we investigated the group GDR(X). Our results show that this group remains very close to the image of Gal(^/A:) in the group of outer automorphisms of the etale fundamental group of X. We shall formulate some conjectures to make this relation precise. Let 7rf*(X x k,x)i 0 Q be the rational Malcev completion of k

Trf*(X x k,x)t/rn+1Tr?(X k

x k,x),. k

Let $ £ : Gal(fe/fe) -> Out (Trf* (X x k.x)^

® Q) be the natural homomor-

AC

phism. Conjecture 15.1. There exists an affine algebraic subgroup Gn of the group Out (Trf*(X xfe,x)\n)®Q) such that $£(Gal(ifc/ik)) is open in Gn(Qz) for the non-archimedean metric ofQi. Let Lie(Gn) be a Lie algebra of the group Gn.

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283

We conjecture furthermore that the Lie algebras Lie(Gn) form a compatible system. We denote the resulting pro-Lie algebra by Lie(<&x(G&l(k/k))). (We hope that such a conjecture has already been made by someone else. We need it to compare the group GDR(X) with the image of Gal(fc/fc). We have formulated the conjecture in analogy with [D4, 8.14].) Let us assume that X is defined over Q. Conjecture 15.2. There is an isomorphism Lie($ x (Gal(Q/Q))) « Lie(gDR(X)) ® Q/. We recall that we have the isomorphism

Let Z((fx ) be thefc-Zariskiclosure of (pxn in ISO(TT?(X(C),z) (n) x fc;7rPR(X,x)(n)). We recall that T{ipxn') is the corresponding torsor (see §11). Question 15.3. Is it true that

Assume that X is a complement of a divisor with normal crossings in a smooth projective scheme of finite type over a number field k and that H})R(V) — Al(V). In this case the group QDR(X) can be considered as a subgroup of Out(npf)). Let CX : G m -> Aut group (n(X)) = Aut Lie algebra(£W) be a homomorphism such that t G Gm(k) acts on H(X) as a multiplication by t. Question 15.4. Does the group 5 D R ( ^ ) contain the image of Cx7> Corollary 15.5. Let V = P^ \ {0,1, oo} and let v = ot be a tangential base point. If Questions 15.3 and 15.4 have affirmative answers for V and v then the transcendence degree of the field Q(27ri)(£(3), C(5), • • • C(2& 4-1)) is k+1. The corollary follows from Theorems A, C and D and Lemma 10.1.3.

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§A.l. Malcev completion. A1.0. Let G be a unipotent algebraic group over a field k of characteristic zero. The exponential map exp : Lie(G) —> G is an isomorphism of Lie(G) equipped with the multiplication given by the Baker-Campbell-Hausdorff formula with G. For a nilpotent Lie algebra L over k equipped with the multiplication given by the B.-C.-H. formula we have Aut groU p(£) = AutLie algebra(£)•

Let R be a fc-algebra. Then the same holds for a unipotent algebraic group scheme over R. A l . l . Let 7T be a finitely generated nilpotent group. Then there exists an affine unipotent algebraic group scheme TTQ over SpecQ and a homomorphism bq : n —> TTQ(Q) functorial with respect to homomorphisms ?r —> TT', which has the following universal property: Let GQ be a unipotent algebraic group over SpecQ. Let

> Gq(Q) be a homomorphism. Then there is a unique morphism of unipotent algebraic groups Gq such that on Q-points we have TTQ(Q) ^^ TTK{K), which is the composition of bq and the inclusion has the same universal property with respect to homomorphisms

GK(K), where GK is a unipotent algebraic group scheme over K. Proof. It follows from Al.l that there is a unique / : TTQ(Q) -» GK(K) such that / o 6Q =
The group TTK together with the homomorphism &K : n —)• TTK(K) has the same universal property with respect to homomorphisms (p : n -> GK(K), where GK is a pro-unipotent pro-algebraic group scheme over K. We call the pair (TTK, 6x : TT —> TTK(K)) the Malcev K-completion of TT. It is unique up to a unique isomorphism. A1.4. It follows from [Ch2] (Theorems 2.1.1 and 2.7.2) and [Q] (Lemma 3.3 and (1.4)) that the pair (af *(V,v), bDR : ^ ( ^ ( C ) , v) -> *f *(V, v)(Q) (resp. (Trf (^(C),t;), b^ : m(V(C),v) -+ Trf (V(Q,v)(Q))) is the Malcev C-completion (resp. rational completion) of ?ri(y(C), v).

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§A.2. The torsor and the group corresponding to the DrinfeldIhara relation. A.2.0. In this section we describe the torsor and the group which correspond to Z/2, Z/3 and Z/5-relations. It is interesting to observe that the Z/3-relation for the group is different from the Z/3-relation in the Galois theory. Let us set X = P 1 (C) \ {0, l,oo}. Let So, Si and SQO be the generators of TTI(X, 01) (see the introduction, and more particularly the picture on p. 5). Observe that S^ • Si • So = 1. Notation. If a and b are elements of the group G we set ab := b • a • b"1. A.2.1. For any Q-algebra k, let us set: TG(k) := {/ : m(X, Of) -> 7r(X)(k)\3t G k* and a(X,Y) G nf(X)(k) with f(S0) = -tX f(Si) = a(X,Y).(-tY)-a(X,Y)-1 *oc) /(5oo) - ( - ^ r 1

• a(X, Z) • (-tZ) • a(X, Z)"1 • ( |

*3) ( - V ) -a{Y,Z) • (-i*Z) -a(Z, Jf) • (-^X) -a(X,y) = 0, *5) a(Xi 2 ,Xi 5 ) • a(X 34 ,X 23 ) • a(X 15 ,X 45 ) • a(X 23 ,X 12 ) • a(X 45 ,X 34 ) = where the last equality takes place in the group 7r(y^)(k). The condition *3 is equivalent to the condition *g) obtained by replacing A.2.2. We shall define an affine group scheme G over Q in the following way. For a Q-algebra k we set: G(k) := ig G Aut(7r(X)(A;))|3s G A;* and a(X,Y) G nf(X)(k) with g(X) = 5X **oo) g(Z) =

**3) a(y, Z) • a(Z, X) • a(X, y ) = 0 **5) a(Xi2,Xi5)

- a(X 34 ,X 23 ) • a(Xi 5 ,X 4 5 ) • a(X 2 3 ,X i 2 ) • a(X 4 5 ,X 3 4 ) = 0.

The last equality takes place in the group 7r(^

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Lemma A.2.2.1. Let us suppose that a(X,Y) andA(X,Y) are inn'(X)(k) and s G A;*. Then there is a unique element (3(X,Y) G n'(X)(k) such that a(sX, (sY)^x'Y^)

• (3(X, Y) = A(X, Y).

Proof. From the equation one gets induction formulas for the coefficients of0(X,Y). 0 Proposition A.2.2.2. The set G(k) is a group. Proof. Let g(Y) = (sY)a^x^ and^i(F) = (tY)^x^Yl ThenagiOg(X,Y) = XiY a(tX, (tY)^ ^)-P(X, Y). It is immediate to check the conditions ** oo , **2 and **3 for agiOg(X,Y). To show that relation **5 holds, we must use the group 7r(3^5)- Let g G G(k). We shall extend g to an automorphism of ft(y5(k) in the following way. We set g(X12) = sX12, g(XM) - sX34, g(X23) = {sx23)«(x^x^x^x™\

g(X45) = g(X51) =

Applying the automorphism g of 7r(3^5)(fc) to the relation **5 for g we get the relation **5 for g\ o g. Assume that g\og = id. Then the element /3(X, Y) is uniquely determined by the equation

(see Lemma A.2.2.1). Applying gi to the equality a(X, Y) • a(Y, X) = 0, we get

a(-Y,, ((-X)^'^-1) •ftx,Yy1 = 0. s

s s

It follows from Lemma A.2.2.1 that

One proves the conditions **oo, **3 and **5 for (3(X, Y) similarly.



A.2.3. We shall show that TG is a G-torsor. Proposition A.2.3.1. The set TG(k) is a G(k)-torsor, i.e. the group G(k) acts freely and transitively on TG(k). If / G TG(k) and g G G(k) then the verification that go f G TG(k) is instantaneous. The action of G(k) on TG(k) is obviously free.

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Suppose that / G TG(k), /(Si) = (-tY)<x^ and h e TG(Jfe), ft(Si) = ). We are looking for g <E G(k) such that g o f = h. If s(y) = then a(X, Y) is determined uniquely by the equation a((pX, ( 0 Y ) a ( x ' r ) ) • a{X, Y) = b(X, Y)

A2.3.2

where (j) = | . Suppose that we have already shown the conditions **oo and **2 for a(X, y). Applying the automorphism g to the formula *3 for / we get a(<j)Y, (4>Z)0(Y>X)'MX>Z)) • (3(Y, X) - (3(X, Z) = 6(y, Z).

Replacing X, 7 by F and Z in A.2.3.2 and using Lemma A.2.2.1 we get /3(y, Z) • /3(Z, X) - /3(X, Y) = 0. The proofs of *oo, *2 and *5 are similar. 0 Proposition A.2.3.3. The functors TG : Q — algebras -* Sets and G : Q — algebras —>• Groups

are representable by an affine pro-scheme over Q and an affine pro-group scheme over Q. We denote these pro-schemes by TG and G respectively. The pro-scheme TG is a G-torsor. Proof. This follows from the definitions of the functors TG and G, Proposition A.2.2.2 and Proposition A.2.3.1. 0 One can show that the monodromy representation #_A : n\ (X, 01) -> TT(X)(C) is an element of TG(C). One takes t = 2ni and a(X, Y) = loga^i(X, Y). This implies the following result.

Corollary A.2.3.4.

References [A]

K. Aomoto, Special Values of Hyperlogarithms and Linear Difference Schemes, Illinois J. of Math. 34 (2) (1990), 191-216. [An] Y.Andre, G-Functions and Geometry, Aspects of Mathematics, F. Vieweg & Sohn Publ., 1989. [C] P.Cartier, Developpements recents sur les groupes de tresses, applications a la topologie et a l'algebre, Seminaire Bourbaki, 1989-90, no. 716, Asterisque 189-190 (1990), SMF Publ., 17-67.

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[Chi] K. T. Chen, Algebra of iterated path integrals and fundamental groups, Transactions of the AMS 156, (1971), 359 - 379. [Ch2] K. T. Chen, Extent ion of C°° function algebra by integrals and Malcev completion of TTI, Adv. in Math. 23 (1977), 181 -210. [Dl]

P. Deligne, Theorie de Hodge II, Publ. Math. IHES 40, (1971), 5 58.

[D2]

P. Deligne, Hodge Cycles on Abelian Varieties, Lecture Notes in Math. 900, Springer-Verlag 1982, 9 - 100.

[D3]

P. Deligne, Equations differentielles a points singuliers Reguliers, Lecture Notes in Math. 163, Springer Verlag, 1970.

[D4]

P. Deligne, Le groupe fondamental de la droite projective moins trois points, in Galois Groups over Q, Math. Sc. Res. Ins. Publ. 16, 1989, 79 - 297.

[D5]

P. Deligne, letter to Bloch, 2.02.1984.

[Dr]

W. G. Drinfeld, On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(Q/Q), Leningrad Math. J. 2 (4), (1991), 829 -860.

[G]

A.Grothendieck, On the De Rham cohomology of algebraic varieties, Math. Publ. I.H.E.S. 29 (1966), 95-103.

[H]

R.M.Hain, On a generalization of Hilbert's 21st problem, Ann. Scient. Ec. Norm. Sup. 4 e serie 19 ( 1986), 609-627.

[HZ]

R.M.Hain, S.Zucker, A guide to unipotent variations of mixed Hodge structure, Lecture Notes in Math. 1246, Springer Verlag, 92 - 106.

[KO] N. Katz and T. Oda, On the differentiation of De Rham cohomology classes with respect to parameters, J. Math. Kyoto Univ. 8 (1968), 199 - 213. [II]

Y. Ihara, Automorphisms of pure sphere braid groups and Galois representations, in The Grothendieck Festschrift, Volume II, Birkhauser, 1990, 353 - 373.

[12]

Y. Ihara, Braids, Galois groups, and some arithmetic functions, Proc. of the ICM 1990 (vol. I) (1991), 99 - 120.

[13]

Y. Ihara, Profinite braid groups, Galois representations and complex multiplications, Annals of Math. 123 (1986), 43 - 106.

[L]

A.Lichnerowicz, Theorie globale des connexions et des groupes d'holonomie, Edizioni Cremonese Roma, 1955.

[MKS] W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory, Pure and Applied Mathematics, XIII, Interscience Publ., 1966.

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[Wl]

[W2] [W3]

[W4] [W5]

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R. Ree, Lie elements and an algebra associated with shuffles, Annals of Mathematics 68(2) (1958), 210 - 220. T. A. Springer, Galois cohomology of linear algebraic groups, in Algebraic Groups and Discontinuous Subgroups, Proc. of Symp. in Pure Math. IX, AMS (1966), 149 - 158. Z. Wojtkowiak, Cosimplicial objects in algebraic geometry, in Algebraic K-theory and Algebraic Topology, Kluwer Academic Publishers, 1993, pp. 287 - 327. Z. Wojtkowiak, Monodromy of polylogarithms and cosimplicial spaces, preprint I.H.E.S., 1991. Z. Wojtkowiak, A note on the monodromy representation of the canonical unipotent connection on P 1 (C)\{ai,..., a n }, preprint MPI, Bonn, 1990. Z. Wojtkowiak, Functional equations of iterated integrals with regular singularities, Nagoya Math. J. 142 (1996), 145-159. Z. Wojtkowiak, Non-abelian unipotent periods, monodromy of iterated integrals, RIMS-969, February 1994 preprint.

Universite de Nice-Sophia Antipolis Departement de Mathematiques Laboratoire Jean Alexandre Dieudonne U.R.A. au C.N.R.S., No 168 Pare Valrose - B.P.N° 71 06108 Nice Cedex 2, France

Research Institute for Mathematical Sciences Kyoto University Kitashirakawa, Sakyo-ku, Kyoto 606, Japan

Part IV. Universal Teichmiiller theory

The universal Ptolemy group and its completions Robert Penner Sur Pisomorphisme du groupe de Richard Thompson avec le groupe de Ptolemee Michel Imbert The universal Ptolemy-Teichmiiller groupoid Pierre Lochak and Leila Schneps

The Universal Ptolemy Group and Its Completions Robert C. Penner* §0. Introduction A new model of a universal Teichmiiller space was introduced in [P2], and a universal analogue of the mapping class groups, called the "universal Ptolemy group", was denned and studied. In our concentration on the geometry in [P2], we perhaps obfuscated the essentially easy algebraic and combinatorial arguments underlying this definition, and one goal here is to give a gentle survey of these aspects and a complete definition of the universal Ptolemy group. To be sure, the geometric side of the story provides the main calculational tools of the theory (and explains the terminology "Ptolemy group"), but we do not discuss these aspects here. To hopefully gain some perspective on the results in this paper, we next review the "universal (decorated) Teichmiiller theory" from [P2]. In this context, the appellate "universal" means that we seek certain infinitedimensional spaces (actually, we shall find infinite-dimensional symplectic Frechet manifolds supporting a group action) together with maps of each of the "classical (decorated) Teichmiiller spaces" into our universal object in such a way that classical combinatorial, topological, and geometric structures arise as the pull-back or restriction of corresponding (group invariant) structures on the universal objects. The classical "(decorated) Teichmiiller theory" may be described as follows. Fix a smooth oriented surface F* of genus g > 0 with s > 0 punctures or distinguished points, and consider the corresponding Teichmiiller space Tg consisting of all complete finite-area metrics of constant Gauss curvature -1 on Fg modulo push-forward by diffeomorphisms of Fg which are homotopic to the identity. The moduli space A4sg is likewise defined to be the collection of all such metrics modulo push-forward by all orientationpreserving diffeomorphisms of Fg. The mapping class group MCsg consists of all homotopy classes of orientation-preserving diffeomorphisms of Fg, and MCsg thus acts on Tgs with quotient Msg. In case s > 1, we may consider the decorated Teichmiiller space Tg which is by definition the total space of the R+-bundle Tg -» Tgs', where the fiber over a point of Tg is the set of all s-tuples of horocycles, one horocycle about each puncture of F*. MCsg evidently also acts on Tgs by simultaneously Partially supported by the National Science Foundation

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pushing-forward metrics and permuting horocycles, and the quotient of Tg by this MG*-action is the decorated moduli space Mg. The sequel does not actually require any real mastery of these classical constructions, and we have included the precise definitions here only for completeness (see [PI] for further details and references) and to describe the following classical square of Fg

fs '9

Msg

> Ts

r

'9

— > Msg

where each morphism in the square is just a corresponding forgetful map, so MCg acts transitively on the fibers of the vertical maps. We shall study here a corresponding universal square Tess

—>

Tess

Mod

—> Mod

relating our corresponding universal spaces and (equally to the point) define a countable group G, the "universal Ptolemy group", one of whose completions acts transitively on the fibers of the vertical maps. There are in fact several reasonable completions of G as we shall discuss. Indeed, it seems clear (cf. [LS] in this volume) that the Galois group of the algebraic closure of Q over Q (to be denoted here simply Galois) arises as a group of automorphisms of a suitable completion of G explaining our exposition of this material here. To abide by our stated goal to gently survey, we shall only highlight various aspects of the theory restricting attention to the righthand side Tess —>• Mod of the universal square for simplicity. This is already sufficient to describe the group G, which is our primary focus in this paper. We define the spaces Tess and Mod in §1 and then define and discuss the group G in §2 taking this opportunity to describe a few recent related results of Imbert-Kontsevich and Lochak-Schneps. §3 and §4 contain complete proofs of several new results; specifically, §3 describes a natural completion of G which acts transitively on the fibers of the map Tess -> Mod and improves our discussion of this in [P2;§4], while §4 describes some related transitivity results which give a new structure theorem for normalizers of Fuchsian groups and suggest still other completions of G as we shall discuss.

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It is a pleasure to acknowledge helpful conversations on various aspects of this work with Bob Guralnick, Pierre Lochak, Leila Schneps, Vlad Sergiescu, and Dennis Sullivan. §1. Tesselations A tesselation r = {ji} is a countable and locally-finite family of geodesies 7»> i > 1> m the Poincare disk D so that each component of D — Ur is an ideal triangle in D. More explicitly, each ji is a bi-inifinite geodesic with well-defined endpoints on the circle S^ at infinity, the ji are pairwise disjoint and decompose D into triangles whose vertices lie at infinity, and this decomposition of D is furthermore locally-finite in D. We refer to a choice of edge ji together with a specification of orientation on it as a "distinguished oriented edge" or a "doe" for short, and we typically denote a tesselation r together with a doe by r1. As a further point of notation, we let r° C SIQ denote the collection of vertices at infinity of the triangles complementary to r, so r° is some countable dense subset of the circle at infinity. There is a standard classical tesselation r#, called the Farey tesselation and illustrated in Figure 1, defined by reflections about the sides of a fixed ideal triangle in D. Explicitly, we have labeled three of our vertices 0 = j , 1 = Y, oo = ~, and these determine a suitable corresponding triangle complementary to T* . We specify that the doe is the oriented edge running from Y to ^ (i.e., a diameter of D) to determine the standard Farey tesselation T' with doe. -2/3

-1/3

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As is well-known [Ra], iteratively labeling the ideal vertices r* of r* by , ^ i 4 £ ^ as illustrated in Figure 1 gives a bijective enumeration of the rationals. (Indeed, the problem of giving a one-to-one enumeration of the rationals was first solved in this manner by the mineralogist Farey without proofs, which were essentially immediately supplied by Cauchy.) Indeed, we say that §? "i? 5?—"i a r e the "first generation" of the enumeration, and each application of the addition law above increases the "generation" by one. For instance, ±^ and ± | are the generation two points. Thus, T® is canonically identified with the set of rational points in the circle at infinity which comes equipped with its Farey enumeration. Furthermore, the Farey enumeration of rf also determines a one-to-one enumeration of r* itself by Q — {+1,-1} as illustrated in Figure 1; namely, given a triangle t complementary to r*, one edge of £, call this edge e, separates the other two edges of t from the doe, and we label e by the Farey number of the vertex opposite e in t (letting 0 denote the doe of Farey by convention). We shall require this enumeration in §3 below. Now, if T' is any tesselation with doe, then there is a unique mapping fr' : r * -* T° defined iteratively as follows. The respective initial and terminal points of the doe of r^ (that is, the points j and ^) are mapped by fT' to the corresponding endpoints of the doe of r'. Next, map the vertex of the triangle to the right of the doe in r* to the vertex of the corresponding triangle to the right of the doe in r'; continue in this way a generation at a time to iteratively define the function / T /. Not only is fT> evidently a bijection between the dense sets T® and r°, but it is also order-preserving (for the counter-clockwise ordering on S^) by construction. It follows from elementary topology and one-dimensional dynamics that fTi interpolates a homeomorphism of 5 ^ which maps T* to r°; this homeomorphism of the circle is also denoted fT> and is called the characteristic mapping of r'. Formalizing part of the discussion above, we define 2

Tess' = {tesselations of D with doe}, Homeo+ = {orientation — preserving homeomorphisms of the circle S^}, so the assignment of characteristic mapping to a tesselation with doe induces a function Tess' —> Homeo+ T'

-> fr'

Conversely, given a homeomorphism / of the circle, we may define a corresponding tesselation f(rf) with doe as follows. Each edge 7 of the Farey tesselation is determined by a pair {rr, y} of points in the circle, there is an associated "image" geodesic f(j) in D spanned by {/(#), /(
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take the "image" f(r') = {/(7) = 7 £ r . } , with doe given by the edge from /(y) to / ( ^ ) . It is straight-forward to check that f(rf) is actually a tesselation and that the assignment / — i > f(rr) is the inverse of the characteristic mapping. We have proved Theorem 1. The characteristic mapping r' — t > fTi induces a bisection between Tess1 and Homeo+.

There are immediately a myriad open questions, for instance, specify some family of orientation-preserving homeomorphisms of the circle and characterize the corresponding family of tesselations. Now, the Mobius group Mob (i.e., the group of orientation-preserving isometries of D, a group isomorphic to PSL(2, R)) acts on the left on both Tess' and Homeo+. We finally define the universal Teichmuller space Tess — Tess' IMob = Homeo+/Mdb, where the topology is induced as the quotient of the pull-back by the characteristic mapping of the compact-open topology on Homeo+. Since the Mobius group acts transitively on ordered triples of points in the circle, we may simply consider "normalized" representatives of Mobiusorbits of tesselations, namely, tesselations where the doe is the edge from j to ^ and the triangle to the right of it is the one spanned by j , y, ^; the corresponding "normalized" homeomorphisms of the circle evidently fix each of these points. In fact, it is easy to give global coordinates on Tess as follows. Given a tesselation r', let us specify an edge 7 of the Farey tesselation, so there is then a corresponding edge of r', namely the image 6 = fr'ii)- Insofar as T' is a tesselation, 5 separates two ideal triangles complementary to r, which together determine an ideal quadrilateral. One of the vertices of this quadrilateral, call it x, is distinguished in that 7 separates it from the triangle to the right of the doe, so we may enumerate the vertices starting from the distinguished one in counter-clockwise order around the circle as x,a,b,c. Recall that the cross ratio of these points is the value x —a

b— c

x —c

b—a

of x under the fractional linear transformation mapping a,b,c H-> 0,1,00, and this is a complete R_ -valued invariant of Mobius-orbits of ordered fourtuples of points in the circle.

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Assigning in this way a cross ratio of points in r° to each edge of Farey descends to a mapping E : Tess -> (R_)T* by Mobius invariance of cross ratios. Furthermore, since cross ratios are a complete invariant, we conclude that E is an injection. In fact, we have Theorem 2. Giving (R_)T* the product topology, E is an embedding onto a path connected open set, so Tess has naturally the structure of a Frechet space. In fact, it turns out that Tess also has a canonical formal symplectic structure as we shall amplify momentarily. Insofar as Tess has been identified with Homeo+/Mob, we can compare it with other such models, namely, with Bers' universal Teichmuller space HomeOqs/Mob, where Homeoqs denotes the space of quasi-symmetric homeomorphisms of the circle (that is, boundary values of quasi-conformal homeomorphisms of the disk D), and with the L2 universal Tecihmuller space Diff+/M6b, where Diff+ denotes the space of orientation-preserving diffeomorphisms of the circle (and the quotients are again by the left action of Mob). We therefore have the inclusions Diff+/Mbb -¥ HomeoqslMb'b -> Homeo+/Mob = Tess, so we are naturally generalizing the other well-known universal Teichmuller spaces; we mention that the spaces in this chain of inclusions are respectively a Hilbert, Banach, and Frechet space. In fact, these inclusions are also geometrically natural, for instance in the sense that our formal symplectic form on Tess pulls back under these inclusions to the Kirillov-Kostant twoform on Diff+/Mob. It follows that our formal symplectic form gives an honest (i.e., convergent) symplectic form on the subspace of all C*+e homeomorphisms of the circle. To complete our description of the right hand side of the universal square, we must still define our model Mod of a universal moduli space, and we shall be content here to describe this simply as a set. As before, we first define the set Mod1 to be the collection of all countable dense subsets of the circle together with the specification of three labeled points a, 6, c among the countable set. The Mobius group acts on the left on Mod', and we let Mod = Mod'/Mob denote the set of Mob orbits. As before, by taking the points a, 6, c to be Y? 1? ^? we may consider "normalized" countable dense subsets. The mapping r' \-+ r° which furthermore takes a, b, respectively, to the initial, terminal point of the doe and c to be the other vertex of the triangle

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in T' to the right of the doe descends to a map Tess —> Mod,

which defines the morphism on the righthand side of the universal square. To see that this morphism is a surjection, fix some normalized countable dense set, and enumerate it xi,X2,x3,..., where we assume that £i,£2,#3 agree with y, y, ^. At each generation of the Farey enumeration, map the new vertices to the least index possible. Specifically, at a given generation of the Farey enumeration, the circle has been decomposed into afinitecollection of open intervals. In each interval, choose the Xi with i least among points lying in the interval and map the next generation of vertices of Farey to these points in each interval. This determines a tesselation r with r° the given countable dense set, as desired. In fact, the quotient topology on Mod is not Hausdorff. Indeed, let us take the rationals r^ and choose one irrational point s in the circle. Define a normalized sequence of tesselations r[ which agree with the Farey tesselation up to some generation z, include an extra vertex at s, and then continue exhausting all the rationals. Each of these tesselations has rf = T£ U {S} by construction yet the limit in i converges to the Farey tesselation, so the two normalized countable dense sets r° and r* U {s} cannot be separated. With a little more work, one similarly sees that the largest Hausdorff quotient of Mod is a singleton. §2. The Ptolemy Group We shall define the group G discussed in the introduction in three stages: groupoid, monoid, and finally group. To define the groupoid, suppose that r' is a tesselation with doe and that 7 € T is an edge of r. 7 separates two triangles in D — r, which together with 7 comprise an ideal quadrilateral of which 7 is a diagonal; let 7* denote the other diagonal of this quadrilateral. We may alter r to produce a new tesselation T7 = (rU{ 7 *})-{7}

as in Figure 2a. We say that r 7 arises from r by applying an "elementary move" along 7 G r. Notice that there is a canonical identification of r — {7} with r 7 — {7*}, so it makes sense to stipulate that if 7 is not the doe of r', then r 7 inherits the given doe of r ' as its doe to determine r 7 . On the other hand, if 7 is the doe of r ; , then we specify 7* as the distinguished edge of ^ taken with an orientation determined by the condition that 7, 7* (in this order) meet with a positive orientation in D. Thus, the elementary

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move on the doe has order four; see Figure 2b. For later use, two further sequences of moves are given in Figures 2c and 2d.

(a)

0-H-0-H-0 (b)

(c)

(d)

Figure 2 Define the groupoid Pt' as a category, where the objects are elements of Tess1 and a morphism from a' to rf is a finite sequence a1 = TQ, T{, . . . , r^ +1 = rf of tesselations with doe, where r/ +1 arises from r[ by an elementary move along some edge of r^, for i — 0, . . . , n . Notice that if there is a morphism between a1 and r', then <J° = r°. Passing to Mo6-orbits, we find a category, called the universal Ptolemy groupoid Pt, whose objects are given by elements of Tess and whose morphisms are given by Mo6-orbits of sequences of tesselations with doe differing by elementary moves as above. To pass to a monoid, observe that given any r' G Tess', the characteristic mapping establishes a bijection between the edges r* of Farey and the edges of r. To specify an elementary move along an edge of r', we might instead simply specify the corresponding edge of r^. Thus, if q £ r*, then we let q • r' denote the tesselation with doe gotten from r' by applying the

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elementary move along the edge fT* (q) of r1 corresponding to q £ r* via the characteristic mapping. Define the free monoid M generated by r*, and inductively extend the action in the previous paragraph by setting ten ' • * 4Mi) * T' = qn • ( . . . • tf2 • tel • T') . . . ) , thereby denning an action of M on Tess'. Passing to Mo&-orbits, we find an action of M on Tess as well. This replacement of a groupoid by a monoid is a general categorical construction applicable whenever the morphisms from any object are a priori identified with some fixed set. To finally define the group, consider the sub-monoid K of M consisting of those words which act identically on r^, that is, whose corresponding groupoid elements begin and end at the Farey tesselation r^ with its fixed doe. For instance, the fourth power of the doe of Farey lies in if as in Figure 2b. The universal Ptolemy group is the quotient G = M/K, where we take the quotient of M generated by insertion and deletion of words in K. To see that G is actually a group, it remains only to verify that there are inverses, namely, that K is big enough to allow inverses in G. For instance, the third power of the doe corresponds to the inverse of the doe in G. In the remaining case of an edge other than the doe, return to the earlier notation of a quadrilateral with diagonals 7 and 7* as in Figure 2a; the word in M corresponding to the composition of the move on 7 with the move on 7* lies in K and provides the required inverse in G. This defines the group G, which we next identify with two other known groups. To this end, suppose that g £ G and consider applying the corresponding elementary moves to the Farey tesselation r^ with its doe to produce another tesselation g • r* with doe. There is a corresponding characteristic mapping fg = fg.T^ giving a representation G —> Homeo+ 9^

fg

of G which is faithful by definition (of the relations K). We mention here that there are also several other interesting essentially geometric representations of G (partly discussed in [P2;p. 180-181] and to be taken up further elsewhere). One can compute directly that in fact the homeomorphism fg actually lies in the subgroup PPSL(2, Z) of piecewise PSL(2, Z) homeomorphisms of the circle. Specifically, if A e PPS r L(2,Z), then there is a finite decomposition of the circle into intervals with rational endpoints, and on each such interval

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/, the function A restricts to an element A\j of PSL(2, Z). Of course, there are conditions on the restrictions A\j to guarantee that they combine to give a homeomorphism, and in fact, this homeomorphism is then automatically once continuously differentiate (cf. [P2; p. 206-207] and [P2;§3], where various properties of PPSL{2, Z) homeomorphisms are discussed). In a recent DEA memoire [II] and also in Imbert's contribution [12] to this volume, we find the following result: Theorem 3. [Imbert-Kontsevich] The assignment g \-t fg is an isomorphism between the groups G and PPSL(2,Z). In fact, there is yet another well-studied group (see [12] and the references therein for a survey) isomorphic to G = PPSL(2, Z), namely, the Thompson group Th defined to be the group of piecewise linear homeomorphisms of the circle where the "breakpoints" (in the piecewise structure) are required to be rational numbers of the form p/2n.

Figure 3 The connection between Th and our current discussion is as follows. Consider the dyadic tesselation r 2 defined a generation at a time as with Farey, but where at each generation, we take the (Euclidean) midpoint of the interval as the next vertex; see Figure 3. With the usual doe (from y to jjj) on r 2 , there is a characteristic mapping / : r'2 —> T£. This homeomorphism evidently conjugates Th to PPSX(2,Z) in Homeo+, and so we find that furthermore G = PPSL(2,Z) = Th. Using Theorem 3 and a known presentation of Th, [LS] have given a new

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essentially geometric presentation of these groups as follows. Corollary 4. [Lochak-Schneps] The groups G, PPSL(2, Z) and Th admit a presentation with relations a\ /33, (a/3)5, [/?a/3, a2papa2], [fia/3, a2$a2pa$a201

a\

where [•, •] denotes commutator and a = the elementary move on the doe (3 — the first six consecutive frames of Figure 2c where • is the doe. In other words, the stated relations generate the sub-monoid K discussed above. To explain the presentation, observe that (3 just cyclically permutes the doe around the base triangle hence has order 3, while a as the move on the doe has order 4. The word a(3 generates the well-known pentagon relation and has order 5. The remaining two relations correspond respectively to the commutativity of elementary moves on disjoint but contiguous quadrilaterals, and quadrilaterals which are separated by one triangle (cf. Figure 2d). Another consequence of Theorem 3 is that the group G is simple (since Th is known to be so), and we therefore cannot simply take the profmite completion of G (there is none!) as a candidate completion to support the desired action of Galois. [LS] describes a certain extension of G by braid groups which does indeed support an action of Galois (where the pentagon relation in G corresponds to the pentagon relation in Galois). In the remaining sections of this paper, we briefly discuss other sensible completions of G which would seem suitable in various contexts but which are not yet known to support any action of Galois. §3. The Transitive Completion In analogy to the mapping class groups, we seek a completion of G which acts transitively on the fibers of the map Tess —y Mod. In fact, in [P2] we gave a a sort of completion (to be recalled below) of G, but it was a set-theoretic completion, not a group-theoretic one. We also proposed there a group-theoretic completion G, which was a "huge" group based on "nets of countable ordinals". We find here that the construction in [P2] is already essentially enough to give a suitable group-theoretic completion G of G based on a much smaller index set. We refer to G as the "transitive completion" since it has the

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required transitivity property. Furthermore, we go on to identify G with a natural group of homeomorphisms of the circle. To begin, we recall some material from [P2;§4.2]. We consider semiinfinite words w = - - - q2q\ where qi G r*, and we read these words from right to left as a corresponding sequence of elementary moves as before. Furthermore, using the identification of r* with Q — {+1, —1} discussed in §1, we regard it; as a sequence of rationals. We must impose two conditions, and the first one is: w = - - q2qi is stable if for each n > 1, there is some L > 1 so that if £ > L, then the Farey generation of the rational qi is at least n; that is, the generation of qt tends uniformly to infinity in £. Given a stable semi-infinite word w = • • • q2qi and r' G Tess1', recursively define r j + 1 = qj+i • rj, where T'Q = r', and let fj denote the characteristic mapping of rj, for j > 0. Let Pn C D denote the ideal polygon of 2 n + 1 sides whose vertices comprise the set P% of Farey points of generation at most n and let P* denote the set of geodesies in T* lying in Pn. Since w is stable, for each n > 1, there is some j n so that fj(e) = fjn(e) for each j > jn and each e G P*. Define

and let Qn C D denote the closed convex hull of Q^. By construction, Qn determines an ideal triangulation of Qn which agrees with the restriction of Tj to Qn for each j > j>n, and Qm C Qn for m 1} = r°; that is, the finite sets {Qn} exhaust the countable set r°. We proved in [P2;§4] that stable and convergent semi-infinite words have a well-defined action on Tess' and Tess, and this action is transitive on the fibers of Tess -» Mod. This basic action and transitivity result will be simply assumed here. (Transitivity essentially follows from the Axiom of Choice and a finite transitivity result to the effect that finite sequences of elementary moves act transitively on triangulations with doe of a finite polygon). To define the completion G, first consider the monoid M consisting of all finite concatenations w = wn - - 'W2wi, where each wi is either a finite word in r* or a stable and convergent semi-infinite word in r*. In case vu is a

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subword of w, where u is semi-infinite, we may think of the first letter of v (and notice that both finite and semi-infinite words have a well-defined first letter) as a limit ordinal for the sequence u. Notice that M is the smallest monoid containing both M and the set of stable convergent semi-infinite sequences. It follows from results already mentioned that there is a well-defined action of M on Tess' and Tess, and we as usual denote the action of w G M on T' G Tess' by w • r1'. Define

so if is a sub-monoid of M. Finally, define the transitive completion G of G to be the quotient G = M/K, which is defined to be the monoid of equivalence classes of elements of M, where the equivalence relation is generated by insertion and deletion of (finitely many) elements of K. Theorem 5. The transitive completion G is a group which acts transitively on the fibers of the map Tess -> Mod. Proof. To see that G is a group, it remains only to verify that K is large enough so that every class in M is invertible, and to this end, it suffices to show separately that the class of each element of G and of each stable and convergent semi-infinite word has an inverse. We have already argued above that elements of G have inverses relative to K C K, so suppose finally that w is a stable and convergent semi-infinite word. Since w • r* is a well-defined tesselation rf with doe and r° = r* by convergence, we may apply the transitivity result from [P2;§4] to produce another stable and convergent semi-infinite word u so that u • r' — r*; thus, the class of u is inverse to the class of w, as desired. Finally, transitivity of G follows again from [P2;§4]. q.e.d. It is worth pointing out that it follows from the proof the every equivalence class in G actually admits a representative w which is a single-letter word in M, that is, either w G G or w is a stable and convergent semi-infinite word (so, in a sense, [P2;§4] was the whole story). It is also worth pointing out that this representative may not always be the most desirable; for instance, if we wish to work equivariantly for a Fuchsian group F, then a useful representative w = wn • • • W2W1 takes each Wi to be a F-orbit of elementary moves on the universal cover. We shall discuss these possibilities further in the next section.

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A reasonable paradigm for the construction of G is to consider the monoid m consisting of all finite concatenations of words which are themselves either a finite sequence of natural numbers or a "stable" semi-infinite sequence • • • n2ni of naturals, where a sequence is "stable" if each natural number occurs in it only finitely often. We construct a sub-monoid k of m generated by n2 for each natural number n and all commutators. The quotient group g = rn/k is isomorphic to the operation of symmetric difference on the power set of the natural numbers. Insofar as G acts on Tess1, given w G G, we might consider the characteristic map fw — fw.ri of w - T^. We also define V = {feHomeo+:f(Q)=Q}, certainly a natural subgroup of Homeo+. Theorem 6. The assignment w — i >• fw establishes an isomorphism between the groups G and V. Proof. First consider an element w G G, so w is represented by wn • • • W2W1, where each Wi is either finite or stable and convergent semi-infinite. Convergence is used to verify that the characteristic map of each Wi indeed sends to Q, hence functoriality of characteristic mappings finally proves that

fwev. On the other hand, suppose that / G V. There is a well-defined "image" /(r^) G Tess1 defined as before, so by transitivity of G, there is some word Wf with Wf - r* = f(rl). The assignment / 1—»• Wf induces the inverse homomorphism, as required. q.e.d. §4. Normalizers of Fuchsian Groups Though parts of what we discuss in this section apply in the more general setting of an arbitrary Fuchsian group, we shall restrict ourselves once and for all for simplicity to the case of a torsion-free finite-index subgroup F < PSX(2,Z). The surface Fr = D/F is a finite branched cover of the "modular curve" D/PSX(2, Z), which has the ramification points 0,1,00 of respective ramification indices 2,3,00, and the points of Fr lying over 00 G D/F5L(2,Z) are necessarily punctures of Fr since F < PSX(2,Z) is finite-index. In contrast, the points lying over 0,1 G D/P5L(2,Z) are smooth points of Fp since F is torsion-free. Furthermore, PSX(2, Z) leaves the Farey tesselation r* invariant (cf. [PI; §6]), so r* descends to an ideal triangulation Ap of Fp? that is, a decomposition of Fr m ^o ideal triangles. Indeed, the edges of Ap are naturally

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identified with the set of F-orbits of edges of r*. More generally, if A is any ideal triangulation of Fr, then we may lift A to a tesselation A of D; F again acts on A, and the edges of A are naturally identified with the F-orbits of edges A. Moreover, for any ideal triangulation A of Fr, we have A0 = r°. An edge e of an ideal triangulation A of Fr is of one of the two following types. We may choose a lift e of e to D, and this lift is the common frontier of two triangles £1,^2 complementary to A in D; let q denote the quadrilateral spanned by these two triangles. It may be that £1 7^ £2 (so the projection is an embedding on the interior of q), and in this case, we say that e is locally separating. In the remaining case that t\ — ti-> q projects to a once-punctured monogon in Fr, and e is the geodesic connecting the puncture to the cusp on the boundary; in this case, we say that e is locally non-separating. Given a locally separating edge e of A, we may sensibly perform an elementary move as before on e in Fr to produce another triangulation of FT- In analogy to the construction of the universal Ptolemy group, let us specify some doe e of A in order to enumerate the edges of A by r*: Given a specified doe e G A, choose some lift e G A to get A' G Tessf, so the characteristic map r* -» A induces a surjection r* -> A. F-invariance of A shows that this enumeration is independent of the choice of e lying over e. (This enumeration depends only on the combinatorics of edges lying to the left/right of the doe in Fr.) In particular, the doe of Farey r* gives rise to a canonical doe of Ar, so every Fr comes equipped with a canonical ideal triangulation Ap with doe. We may specify, as before, a sequence of elementary moves on a triangulation A' with doe of Fr instead as a sequence of elements of r*. We must include also letters corresponding to elementary moves on locally nonseparating edges to achieve this a priori enumeration of A, and we simply think of these letters here and throughout as acting identically on triangulations with doe. Thus, the free monoid M on r* acts naturally on the set of triangulations with doe of Fp, and we may build the kernel KY corresponding to all sequences of moves which act identically on Ap to define the Ptolemy group Gp = M/KY of F. Lemma 7. GY is a group. Proof. Again we must simply prove the existence of inverses. To this end, recall from [Pl;Proposition 7.1] that finite sequences of elementary moves in Fp act transitively on ideal triangulations of Fp. Thus, given w G M, we may find u G M so that (uw) • Ap agrees with the tesselation (without doe) underlying Ap. We can finally find a word v G M in the letters a2 and (3

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of Corollary 4 above to arrange that (vuw) • (Ap) = Ap, so the class of vu is inverse to the class of w. q.e.d. In particular, if e € r* corresponds to a locally non-separating edge in Fp, then e lies in Kr, and if ei, e^ € r* lie over a common edge of Fr, then ei = e2 in Gr> In fact, we are more keenly interested here in another group, a certain extension of Gp by F to be discussed below. In case e\ and £2 are edges of A' lying over a locally separating edge e of A, then the elementary moves on e\ and £2 commute as in Figure 2d. Explicitly, we have hAe2)

ei

=

/e a (ei) e2

in G r ,

where, as before, /e;(e?) is the "image" of ej under the characteristic map /e, of ei-r^. Now, suppose that e is a locally separating edge in a triangulation A' of Fp with doe. Enumerate the edges e^, i > 1, of A lying over e regarded as elements of r*, and consider the formal semi-infinite word

Our immediate goal is to show that this is actually a stable and convergent semi-infinite word. Assuming this, the order ei,e2> • • • on the edges lying over e is immaterial according to the commutativity discussed in the previous paragraph; that is, different orders give rise to the same action on tesselations with doe. On the other hand, it is convenient in the sequel to actually specify a particular order as follows. Choose a fundamental domain E G D for F which is a connected ideal polygon triangulated by A, where we may assume that the doe lies in E and some fixed lift e\ of e lies in the interior of E. We say that E itself is "first generation" and also that the vertices of E are "first generation". Of course, the frontier edges of E are identified in pairs by F, and across each frontier edge of E there is therefore some other contiguous fundamental domain; these are the "second generation" fundamental domain, and their vertices taken together form the "second generation" of points in the circle. One continues in this way to associate a generation to each ideal point of A and to each F-orbit of E. Let us choose an enumeration E = E\, E2,... of the F-orbits of E so that the generation of 2?j+i is at least as great as the generation of Ei, for each i > 1. In the interior of each E^ there is a unique lift e^, and we define the T-move to be the semi-infinite word Wq='-

fester (e4) /e2ei (h) Hx (h) ^1,

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where gGr* corresponds to e G A' as above. Lemma 8. For any edge gGr* and any T, w^ is a stable and convergent semi-infinite word. Proof. If the edge q corresponds to a locally non-separating edge, then w^ acts identically and is thus stable and convergent by convention. In case the edge q corresponds to a locally separating edge, then adopt the notation of the previous paragraph. Stability is clear just from the existence of the ordering above on fundamental domains by non-decreasing generation. Since e; lies in the interior of Ei by construction, there is always at least one triangle in Ei with e^ in its frontier and separated from the doe by e^. Thus, the set of points in the circle of generation at most n contains the set of Farey points of generation at most n by induction, so the word is convergent as well. q.e.d. It follows from [P2;§4] and Lemma 8 that for any edge q G r* and any Finvariant tesselation r' with doe of D, there is a well-defined corresponding tesselation with doe wvq • r', and we furthermore have its characteristic map fq,T"

Consider the normalizer NHomeo+(r) = {/ G Borneo^ : / F / " 1 - F} of F in Homeo+. It follows from the defining equation / F / " 1 = F that / necessarily maps the set of parabolic fixed points of F to itself, that is, / necessarily satisfies /(Q) = Q. It follows that / G V, so iVy(F) = NHomeo+ (F), and we let =

NHorneo+(T)

denote the normalizer of F in either group. Lemma 9. For any edge q G r*? any T, and any F-invariant tesselation r' with doe, we have f^T, G N(T). Proof. To see that /£ T , normalizes F, consider two copies of the universal cover of Fp, where F acts on each copy, but we take r' in the first copy and Wq - r' in the second copy. /£ T , induces a topological mapping on the universal covers which sends r' to w^ • r', and so / T r , conjugates F to itself. q.e.d. We next show that conversely any / G iV(F) can be expressed as the characteristic map of a finite composition wn • • • w2wi, where each wi is a

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F-move. The proof again depends upon [PI; Proposition 7.1] as in Lemma 7. One may think of this result as a kind of structure theorem for normalizers of Fuchsian groups. At the same time, these results together may be seen as a sort of converse to Nielsen's famous theorem that a homeomorphism of a surface Fr lifts to a mapping on D which extends continuously to a homeomorphism of S^ normalizing F. Given a F-move w^, where g 6 r*, we may regard the stable and convergent semi-infinite word w^ as acting on r[ itself. Letting M as usual denote the free monoid over T*, an element w — qn • • • <72
Define KY = {w E M : w • r^ = r*} and the F covering Ptolemy group Gr =

MjKv.

One sees that Gr is indeed a group arguing just as in the proof of Lemma 7. Furthermore, Gr acts on the set of F-invariant tesselations with doe as before, and we have an exact sequence 1 -> F -> G r -> G r -> 1. If w € Gr, then we let fw denote the characteristic map of w • r*. Theorem 10. The assignment w — i »• fw establishes an isomorphism between the groups Gp Proof. Using Lemma 9, the mapping is evidently an injective homomorphism. To see that it is surjective, choose a fundamental domain for F, and take its "image" under / as before, for any fixed / in N(T). This is again a fundamental domain for F, as one can check, and it comes equipped with an induced triangulation with doe. Now, as in the proof of Lemma 7, there is a sequence of F-moves taking Farey to its "image" under / and mapping the doe correctly, so surjectivity follows. Finally, use that / normalizes F to prove that this construction is independent of the choice of fundamental domain. q.e.d. Relating the constructions of this section to the transitive completion G discussed in §3, we have Corollary 11. For any V, the assignment M -> M induced by q H-> W^ descends to a canonical injection N(T) = Gr —> G.

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As to completions, there is a homomorphism N(Tt) = GTl -> Gr2 = N(T2) whenever F2 < I \ defined as follows. Given q G r^ and any Fi-invariant tesselation rf with doe, there are induced triangulations with doe A^ and A'2, respectively, of Fr1 and Fp2. Let e G Ai correspond to q G r* and consider the full pre-image of e under the projection Fp2 —> -Frr> choose labels <# G r*, for i = 1,..., n corresponding to this family of arcs in Fr 2 , and define Q*->fqn-i -9291 fan) " • /
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[P2] —, Universal constructions in Teichmiiller theory, Adv. Math. 98 (1993), 143-215. [Ra] H. Rademacher, Lectures on Elementary Number Theory, Blaisdell, New York, 1964. [Su] D. P. Sullivan, "Relating the universalities of Milnor-Thurston, Feigenbaum and Ahlfors-Bers", in the Milnor Festschrift Topological Methods in Modern Mathematics, eds. L. Goldberg and A. Phillips, Publish or Perish (1993), 543-563. Departments of Mathematics and Physics/Astronomy University of Southern California Los Angeles, CA 90089

Sur l'isomorphisme du groupe de Richard Thompson avec le groupe de Ptolemee Michel Imbert

§0. Introduction. Le but de cet article est de donner une demonstration de Fobservation suivante de M. Kontsevitch: le groupe universel de Ptolemee G defini par R. Penner dans [Pel] est isomorphe au groupe de R. Thompson PI^S 1 ). Le groupe PL 2 (S 1 ) a ete decouvert par R.Thompson dans un contexte algebrique et ce sont ses notes manuscrites non publiees qui constituent la reference de base. Le groupe G est lui un ingredient important dans la theorie de Penner des espaces universels de Teichmuller, il possede par exemple des completions interessantes (voir Particle de Penner dans ce volume); d'autre part P. Lochak et L. Schneps ont etendu le groupoide universel de Ptolemee (a partir duquel R. Penner a construit G) en un nouveau groupoide qui possede une completion profinie sur laquelle le groupe de Galois absolu agit en tant que groupe d'automorphisnaes (voir leur article dans ce volume). Nous suivrons le plan suivant: dans une premiere partie nous rappellerons la definition de PL2(51) et quelques formes equivalentes de ce groupe, ce qui nous permettra de demontrer dans une deuxieme partie qu'il est isomorphe a G, et ceci de deux manieres. Dans tout Particle nous noterons Homeo+ le groupe des homeomorphismes du cercle preservant l'orientation, et Q(2) l'ensemble des nombres dyadiques de Tintervalle [0,1], ie les nombres de la forme ^ compris entre 0 et 1 avec q et n entiers naturels. Enfin nous identifierons souvent S1 avec R U {oo} et avec [0,1] ou 0 et 1 sont identifies. Je suis redevable a M. Kontsevitch pour la communication de ce resultat, et je remercie V. Sergiescu, par lequel j'en ai eu connaissance, de m'avoir encourage a le demontrer.

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§1. Le groupe de R.Thompson §1.1. Definition, survey. Definition 1.1. PL^S 1 ) est le sous-groupe de Horrieo+ forme des elements qui sont affines par morceaux et tels que: 1. les points de coupure appartiennent a Q(2). 2. Pimage de 0=1 appartient a Q(2). 3. les derives de chaque application affine constituant un element du groupe sont des puissances de 2. Ce groupe, defini en 1965 par R. Thompson, ainsi que F le groupe des homeomorphismes de [0,1] lineaires par morceaux possedant les proprietes I et 3, fut introduit au debut par son auteur pour construire des groupes de presentation finie avec un probleme de mots non resoluble (voir [M,Th]). II s'est avere ensuite posseder des proprietes remarquables et il est apparu dans de nombreux contextes mathematiques. Voici une liste non exhaustive de ces proprietes: PL2(S'1) est un groupe infini, simple, de presentation finie. C'est un resultat de R. Thompson qui est demontre dans [CFP], article ou Ton trouvera les proprietes basiques de ce groupe. W. Thurston a trouve une interpretation de PL^S 1 ) liee aux structures projectives entieres par morceaux (voir plus bas). Dans [GhS] les auteurs classifient les actions de PILES'1) sur le cercle suivant la dynamique topologique de cette action, et donnent une description de ses representations dans DifFr(51) pour r> 2; d'autre part ils donnent une serie de resultats sur rhomologie de ce groupe (par exemple Hn(PL2(S1), Z) est de type fini pour tout n) ainsi que diverses relations avec les classes caracteristiques. Dans les annees 80, F est egalement apparu en theorie de l'homotopie et en homologie des groupes. En particulier, K.S. Brown et R. Geoghegan ont montre que les groupes F et PL^S 1 ) sont de type FPQQ (voir par exemple [Br]). §1.2. Le groupe modulaire par morceaux. Definition 1.2. PPSL2(Z) est le groupe des homeomorphismes preservant l'orientation de RU {oo} qui sont PSL^Z) par morceaux, avec des points de coupure rationnels. Theoreme 1.1. PPSL2(Z) est isomorphe a PL2(S'1).

Groupe de Thompson

315

Esquisse de la demonstration. Elle repose sur une bijection canonique que nous noterons i entre Qu {00} et Q(2) et qui est construite ainsi. On envoie d'abord 0/1 sur 0 = 1, 1/1 sur 1/4, 00 = 1/0 sur 1/2 et - 1 / 1 sur 3/4. Pour une fraction reduite a/b, appelons max(|a|, |6|) l'ordre de a/b. Alors si q € Q + est d'ordre n, q s'ecrit | ^ ou a/b et c/d sont d'ordre au plus n - 1 (voir [Ch] pages 6 a 8). Soit d\ — i(a/b) et d
est un isomorphisme de groupe.

0

§1.3. Couples d'arbres. Les elements de P I ^ S 1 ) peuvent etre parametres par des couples d'arbres binaires ordonnes munis d'une permutation cyclique des feuilles. Pour ceci la reference est [Br]. Definition 1.3. Un arbre binaire ordonne est un arbre S tel que: 1. S ait un sommet distingue ^0 (appele la racine) tel que si S n'est pas reduit a v0 alors v0 est de valence deux. 2. si v est un sommet de S de valence >2, alors il y a exactement deux aretes ev^\ et ev>r, appelees respectivement arete gauche et arete droite, qui contiennent v et qui ne sont pas contenues dans le chemin d'aretes allant de ^0 au. Les sommets de valence un sont appeles des feuilles et peuvent done etre ordonnes de gauche a droite. Definition 1.4. Un intervalle standard dyadique (i.s.d) est un intervalle ferine inclus dans [0,1] de la forme [^-,^r] ou a et n sont des entiers positifs ou nuls. Une partition de [0,1] est dyadique si et seulement si les intervalles de la partition sont des i.s.d. II existe un arbre infini dyadique note ^4^ dont les sommets sont les i.s.d, et une arete est une paire (/, J) d'i.s.d telle que si J est la moitie gauche (resp. droite) de / alors (7, J) est une arete gauche (resp. droite). Nous appelons ici un arbuste un arbre binaire ordonne fini identifie a un sousarbre binaire ordonne fini de ^loo (i.e. un sous-arbre fini de racine v0 dont les aretes gauches (resp. droites) sont des aretes gauches (resp. droites)

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Michel Imbert

de Aoo). II y a une bijection canonique entre les arbustes et les partitions dyadiques de [0,1], les feuilles correspondants aux intervalles de la partition. Exemple.

Definition 1.5. Un diagramme arboricole est une paire ordonnee d'arbustes (iZ, S) qui ont le meme nombre de feuilles, munie d'une permutation cyclique des feuilles, cr, modulo la relation d'equivalence engendree de la maniere suivante. Soit (R, 5, a) un tel diagramme, / la n-ieme feuille de R et a(I) la feuille correspondante de S. Si on rajoute a R n'importe quel arbre binaire B ordonne a la racine / et si on rajoute a S le meme arbre a la racine cr(/), alors on dira que (i?,5,a) est equivalent a(J?UB,5uB,a') ou a' se deduit de a et de l'identite sur B. Si / £ PL2(5 1 ), alors il existe des partitions dyadiques P et Q telles que / soit affine sur les intervalles de P, et les envoient sur les intervalles de Q (voir lemme 2.2 de [CFP]). On peut done associer a / un diagramme arboricole {R, 5, a) ou R est l'arbuste associe a P et 5 celui qui est associe a Q. Si /(0) = 0 alors a=id. Notons DA l'ensemble des diagrammes arboricoles. Si (Q,R,a) EDA et (/?, T, r) €DA correspondent respectivement a / et g alors (Q,T, rcr) correspond a g o f. Theoreme 1.2. DA est un groupe isomorphe a PILES'1). On en trouve la preuve dans [CFP], ou les auteurs donnent aussi une presentation de DA en terme de trois generateurs et six relations, trouvee par R. Thompson (voir la figure suivante ou ces trois generateurs sont exprimes dans PPSL2 (Z), P I ^ S 1 ) et DA ).

Groupe de Thompson

PPSL

PL (S 1 )

(Z)

z A

1/2

z+1 1/4 0

id z

B

"!

*' Z " z

sur [0/1,1/0] sur [1/0,-1/1]

317

J

Ml 3/4

DA

CM CT = id

1

1 3/4 5/8 1/2

sur [-1/1,-1/2]

/

z sur [-1/2,0/1]

0

1/2

CJ = id

3/4 7/8 1

1 -1

3/4

1+z

1/2

\

2

3

2

J

C

0

1/2

3/4

I

CJ=(132)

§1.4. Couples de polygones triangules. Le groupe DA est isomorphe a un groupe que nous noterons DP qui consiste en des couples de polygones triangules ordonnes munis d'une permutation. II s'agit d'une construction de dualite classique, voir par exemple [STT]. Soit done R un arbuste. A chaque sommet on fait correspondre un triangle; en particulier on fait correspondre a la racine de l'arbuste v0 un triangle de base To. A une arete de R entre deux sommets on fait correspondre une instruction de collage des deux triangles correspondants le long d'une arete. On forme ainsi un polygone triangule P(i?). Grace a la propriete 2 de R, on observe que si T est un triangle de P(R) de valence >2, alors il y a exactement deux triangles ST,I et Sr,r adjacents a T et non contenus dans le chemin de triangles allant de To a T. Les triangles de valence un (i.e. avec deux cotes libres) correspondent aux feuilles de l'arbuste; ils sont dits externes. Ils sont ordonnes de gauche a droite.

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Definition 1.6. DP est l'ensemble des diagrammes polygonaux constitues de couples de polygones triangules ordonnes avec le meme nombre de triangles externes, munis d'une permutation de ces triangles externes, et modulo une relation d'equivalence analogue a celle definie pour les diagrammes arboricoles. On obtient sans difficulty le theoreme suivant: Theoreme 1.3. DP est un groupe isomorphe a DA. Un fait important pour l'isomorphisme entre PL2(51 ) et le groupe universel de Ptolemee est le resultat suivant de transitivite : etant donnes deux polygones triangules a n cotes, il existe une serie finie de mouvements element air es permettant de passer de l'un a l'autre, ou un mouvement elementaire est un changement de diagonale a l'interieur d'un quadrilatere (voir [STT] et lemme 4.4 de [Pel]). §2. Le groupe universel de Ptolemee. §2.1. Tesselations. Le paragraphe 2.1 reprend quelques notions des deux articles de R. Penner cites en references. Definition 2.1. Une tesselation r est une collection denombrable et localement finie de geodesiques du demi-plan de Poincare % dont les regions complementaires sont des triangles ideaux de sommets appartenants a RU {oo}. r(°) denotera les sommets de r. Us sont denses dans Ru{oo}. r^ designe l'ensemble des triangles complementaires de r dans %. II faut considerer des couples (r, e) ou r est une tesselation et e est une geodesique distinguee des autres que Ton oriente. L'ensemble de ces couples est note TESS' par R. Penner. L'avantage de ces couples reside dans le fait qu'ils sont combinatoirement rigides, ie : si (T;, e^) pour i = 1,2 sont deux tels couples et x», yi sont les sommets de e^, alors il existe une unique bijection /

: r

~~^ r 2

i

Xi !->> X2

y\ ^ 2 / 2

et telle que si (x,y,z) engendre un triangle de r{ ^ alors (f(x),f(y),f(z)) engendre un triangle de r^ . / respecte l'ordre et determine une application caracteristique F : T\ —y T2

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Definition 2.2. L'espace universel de Teichmiiller, TESS, du a R. Penner est defini comme les classes d'equivalence des orbites des couples (r, e) de TESS' modulo Faction de PSL2(R). II existe une tesselation particuliere, dite de Farey, qui joue un role important dans la theorie de Penner. On considere le triangle geodesique de sommets 0, 1, oo. La tesselation de Farey consiste en les images de ce triangle par PSL2(Z) . Le groupe modulaire est alors le sous-groupe de PSL2(M) qui preserve cette tesselation. On note r* cette tesselation, et son arete orientee notee e0 est celle qui va de 0 vers oo. Ce qui est remarquable, c'est qu'il y a une bijection entre T;' et Qu{oo} : si T G r* possede deux sommets assignes respectivement a a/b et c/d alors le troisieme sommet est assigne a |q^. Soit maintenant Q7 =\{—1,1}. Alors on a la bijection suivante:

e0

ou eq est definie ainsi: soit e E r* (e / e0 ), U la composante connexe de % \ {e} qui ne contient pas eo, et T l'unique triangle de r* DU ayant e dans sa frontiere. Un des sommets de T n'appartient pas a e; il est assigne a un rationnel q, et alors e = eq. Voir la figure suivante represent ant la tesselation de Farey.

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Lemme 2.1. Homeo+ /PSL2(M) et TESS sont des espaces topologiques homeomorphes. Esquisse de la demonstration. Si g G Homeo+, on lui associe la tesselation (g(e) : e £ r*) ou si x et y sont les sommets de e alors #(e) est l'unique geodesique joignant g(x) hg(y), munie de l'arete orientee distinguee joignant a. #{°°}- D'autre part si (r, e) ETESS', on lui associe l'unique element de Homeo+ qui prolonge / : r*

—» r^0^.

<^

§2.2. Le groupe universel de Ptolemee et l'isomorphisme remarquable. Soit (ri,e) ETESS' et a une arete de TI; soit a' l'autre diagonale du quadrilatere forme par les deux triangles ayant a dans leur frontiere; enfin soit F : r* —>> ri l'application caracteristique et a = F(eqi). Alors r 2 = (ri\(a)) U (a;) est une autre tesselation. On ecrit r 2 = q\ • T\ et on dit que Ton a obtenu r2 par un gi-mouvement. II faut remarquer que r{ ' = r^0 . D'autre part on possede une bijection * i : n -> r2 a h-> a',

et qui vaut l'identite ailleurs. Si l'arete orientee distinguee de T\ est differente de a, alors elle sera aussi celle de r2; sinon a' devient l'arete distinguee, et on l'oriente de telle fagon que {a, a1) soit une base directe du plan. Soit M le monoide libre forme sur (Q/ et soit une action a gauche de M sur TESS donnee par: M x TESS -> TESS (Qi'"Qn = m, [ri]) ^ [m • n ] ou m-ri

= qi(- • • (gn_i(9n • 7i)) • • •)•

Ensuite, si m = ^i • • • q\ et n — q[ • • • ^ € M et si on pose 3>m = $ i o ^>2 • • • o 3>j et n = $i o $2 . •. o <£p, alors m et n sont dits equivalents si et seulement si pour tout [r] 6TESS, on a [ra-r] = [n-r] et les correspondances induites $ m et <£n sont identiques. La classe d'equivalence d e m G M est notee [777,]. Lemme 2.2. G, Vensemble des classes d'equivalence de M, muni de Voperation [m][n] = [mn] est un groupe, dit universel de Ptolemee, qui agit a gauche sur TESS. On trouve la demonstration dans [Pel].

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Voici maintenant le theoreme qui motive cet article, initialement demontre dans [Im]. Theoreme 2.3. G est isomorphe a PPSL2(Z), et par consequent a PL^S 1 ). Demonstration. Demontrons d'abord l'existence d'un homomorphisme injectif de groupes entre G et PPSL 2(Z). Dans [Pel], R. Penner donne une representation fidele de G dans l'ensemble des applications de r* dans r*. J'ai remarque dans [Im] qu'en passant a l'ensemble des sommets ri ' cela fournissait un plongement de G dans Homeo+. Enfin dans son article dans ce volume, R. Penner resume cela ainsi: soit g G G, et fg : T* ' —>• {g-r*)^ = r* 1'application correspondante. Elle se prolonge de maniere unique en un element de Homeo+ qui preserve l'ensemble des nombres rationnels et que Ton note encore par fg. Alors h : G —> Homeo+ 9^

f9

est une representation fidele de G. Je demontre alors que l'image de G est incluse dans le sous-groupe PPSL2(Z) de Homeo+. Soit done q G Q^, qu'on suppose strictement positif, et Q le quadrilatere forme par les deux triangles possedant eq dans leur frontiere. II y a une unique geodesique eq> appartenant a la frontiere de Q qui est dans la composante de % \ Q contenant eo- Si q — c/cf est strictement superieur h q' = b/bf, alors on voit facilement que les sommets de Q sont dans l'ordre a/a', b/b\ c/d, d/d1 avec c = b + d,d = V + d' et b = a + d,V = a1 + d'. Dans la composante de % \ Q qui contient eo, y compris sur e^, on a Fq =id. Dans les trois autres composantes de 1-L\Q, Fq est une bijection respect ant la structure triangulaire d'une partie infinie de r* dans une autre partie infinie de r*. Comme Aut(r*) est le groupe PSL 2(Z), sur chacune de ces trois composantes, y compris sur leur frontiere avec Q, Fq est egal a un element de PSL2(Z). On en deduit que fq est une bijection de R U {CXD} qui se comporte comme un element de PSL^Z) sur chacun des intervalles [a/a',6/6'], [b/b',q], [q,d/d% [d/d',a/a% i.e. fq e PPSL2 (Z). On procede de maniere analogue si q est strictement inferieur a q'. Enfin si q est negatif la demonstration est egalement analogue. Pour prouver que notre homomorphisme est en fait un isomorphisme de groupes, il sufRt de trouver h' : PPSL2(Z) —> G tel que h o h' =id sur PPSL2(Z). Les elements de PPSL2(Z) sans points de coupure sont ceux de PSL2(Z); par contre il n'existe pas d'elements avec un ou deux points de coupure. Soit g e PPSL2(Z) avec au moins trois points de coupure. On lui associe r isi~1) — (^" 1 (e),e G r*). Soient r i , r 2 , . . . ,r p (p > 3) les p points de

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coupure de g~l, et posons Si = g~l(ri) et n = max(ordre(si)) pour i G { 1 , . . . ,p}; soit Pn le poly gone geodesique regulier ouvert dont les sommets sont les rationnels d'ordre au plus n. (L'ordre d'une fraction reduite a/b est egal a max(|a|, |6|).) Les aretes frontieres de Pn et exterieures sont retrouvees intactes dans r{g~x) car Aut(r*) = PSL2 (Z). Par contre, les aretes situees a l'interieur de Pn peuvent etre remplacees par de nouvelles aretes. Grace au lemme 4.4 de [Pel] il existe m(g) G M tel que m(g) • r* = rig-1). Ondefinit ti{g) = [m(g)}. Si g G PSL2(Z), on ecrit g = (goh)oh,-1, goh et h~x possedent alors quatre points de coupure si h en possede quatre, et on definit h'(g) = [m(g o /i)m(ft~1)]. Soit maintenant g G PPSL 2 (Z). Alors hoh'(g) = / , l'unique bijection de rigidite qui envoie l'arete orientee distinguee de r(g~~l) sur eo et les triangles de r(g~1)^ sur ceux de ri . Or par construction g possede ces proprietes done h o hf =id. Dans la pratique, on peut calculer directement fq a partir de q; on calcule ainsi que: sur [0/1,1/1] sur [1/1,1/0] sur [1/0,-1/1] sur [-1/1,0/1]. h([0])2 vaut - l/z.

{

id

sur [1/0,0/1]

z-1

sur [2/1,1/0].

En utilisant ce theoreme et la presentation de PL 2 (5X ) deja citee au §1.3, P. Lochak et L. Schneps ont trouve une presentation remarquable de G en termes de deux generateurs et cinq relations (voir leur article dans ce volume). Ces deux generateurs sont [0] et [0][2/l][0]3[2/l][0] qui correspond au generateur C de la presentation precitee. Corollaire 2.4. L'homeomorphisme entre Homeo+ \ PSL2(R) et TESS est equivariant pour les actions respectives de PPSL2 (Z) et G. De plus PPSL2(Z) (et done G) agit sur Vespace universel de Bers Homeoqs\PSL2(R) oil HomeOqs est le sous-groupe de Homeo+ forme des elements qui sont quasi-symetriques. Demonstration. On a les actions suivantes: PPSL2(Z) x Borneo+ \ PSL2(R) -> Homeo+ \ PSL2(R)

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ou [h] represente une classe a gauche; et G x TESS -> TESS

Comme, d'apres la preuve du theoreme 2.3, on a r{hog~x) = m(g)-T(h), on a bien l'equi variance. La deuxieme partie du corollaire resulte du fait suivant: les elements de PPSL2(Z) sont de classe C1 done quasi-symetriques. 0 On peut voir dans [Pel], a la fin du §1, l'interet de la derniere action du corollaire. Nous allons maintenant donner une deuxieme demonstration du theoreme 2.3, suggeree par V. Sergiescu, en demontrant que G est isomorphe au groupe DP forme de couples de polygones. Demonstration. Soit gGQ' strictement positif; on considere le 2 tel qu'il n'y ait rien de change a l'exterieur du polygone ouvert P n . Soit In le polygone triangule forme des geodesiques de r* reliant les rationnels positifs d'ordre inferieur ou egal a n; et soit Sn le polygone triangule forme des geodesiques de r* reliant les rationnels negatifs d'ordre inferieur ou egal a n. On associe a q Pelement (i?n , Q n , id) de DP, ou Rn et Qn sont construits ainsi : soit T un triangle de base, a et b deux de ses aretes; on obtient Rn en collant T avec In (resp Sn) grace a une instruction de collage de a avec l'arete (0/1,1/0) (resp. de b avec l'arete (0/1,1/0)); enfin on obtient Qn en remplagant In par q- In, le nouveau polygone triangule obtenu apres le g-mouvement. On procede de meme pour q strictement negatif, en remplagant Sn par q • Sn. Enfin si q = 0, on lui associe l'element (i?2,i?2, (1234)). On en deduit une application de M dans DP, et par definition de G et de son action sur les tesselations on a done construit un homomorphisme inject if de groupes entre G et DP. La surjectivite de cet homomorphisme decoule encore du resultat de transitivite finie cite a la fin du §1.4. Soit (P, Q, a) un couple de polygones triangules ordonnes et une permutation de ses triangles externes; alors en inject ant P dans un certain Rn en identifiant le triangle de base avec le triangle T, on sait que Q s'injecte dans le meme polygone que P mais avec une triangulation differente, ce qui permet d'appliquer le lemme de transitivite finie, et en repassant a la tesselation de Farey de voir que (P, Q, a) est 1'image d'un element de G. 0 Remarque. Par definition l'arbre de Farey consiste en deux exemplaires de l'arbre infini dyadique defini au §1.3 relies par une arete joignant les racines. On peut le construire egalement comme dual a la tesselation de Farey. Alors on demontre facilement que le groupe universel de Ptolemee G

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est isomorphe au groupe des germes d'automorphismes a Tinfini de Tarbre de Farey. Cette action a 1'infini sur l'arbre de Farey est utilise dans [GrS] pour construire une extension acyclique du groupe de tresses a partir du groupe F mentionne au debut du §1.1. References [Br] [CFP]

[Ch] [GhS] [GrS] [Im] [M,Th]

[Pel] [STT]

K.S Brown, Finiteness properties of groups, J. Pure App. Alg. 44 (1987), 45-75. J.W. Cannon, W.J. Floyd, W.R. Parry, Notes on Richard Thompson's groups, Preprint (University of Minnesota, 1994), a paraitre dans Ens. Math. K. Chandrasekharan, Introduction to analytic number theory, Springer Verlag (1968). E. Ghys, V. Sergiescu, Sur un groupe remarquable de difTeomorphismes du cercle, Comm. Math. Helv. 62 (1987), 185-239. P. Greenberg, V. Sergiescu, An acyclic extension of the braid group, Comm. Math. Helv. 66 (1991), 109-138. M. Imbert, Constructions universelles dans la theorie des espaces de Teichmiiller, Memoire de DEA, Grenoble (1994). R. McKenzie, R.J. Thompson, An elementary construction of unsolvable word problems in group theory, Word problems, W. W. Boone, F.B Cannonito and R.C Lyndon, eds., Studies in Logic and the Foundation of Mathematics, Vol. 71, Amsterdam (1973), 457478. R. Penner, Universal constructions in Teichmiiller theory, Adv. in Math. 98 (1993), 143-215. D. Sleator, R. Tarjan, W. Thurston, Rotation distance, triangulations, and hyperbolic geometry, J. of the Amer. Math. Soc. (1988), 647-681.

The universal Ptolemy-Teichmiiller groupoid Pierre Lochak and Leila Schneps

Abstract We define the universal Ptolemy-Teichmiiller groupoid, a generalization of Penner's universal Ptolemy groupoid, on which the Grothendieck-Teichmiiller group - and thus also the absolute Galois group - acts naturally as automorphism group. The essential new ingredient added to the definition of the universal Ptolemy groupoid is the profinite local group of pure ribbon braids of each tesselation.

§0. Introduction The goal of this article is to give a completion (by braids) of Penner's Ptolemy group G such that there is a natural action of the GrothendieckTeichmiiller group (and a fortiori, the absolute Galois group Gal(Q/Q)) on it. This work was motivated on the one hand by the deep relation of the Ptolemy group - shown to be isomorphic to Richard Thompson's group and to the group of piecewise PSL2(Z)-transformations of the circle - and mapping class groups and the geometry of moduli spaces in general, most visibly in genus zero, and on the other by the presence of the remarkable pentagonal relation, stimulating the natural reflex of the authors to associate every pentagon appearing in nature to that of the Grothendieck-Teichmiiller group GT. The difficulties in defining a GT-action on G were the following. Firstly, the profinite version of GT which interests us (mainly by virtue of the fact that it contains the Galois group) acts on profinite groups, whereas via its isomorphism with Richard Thompson's group, G is known to be simple, and therefore its profinite completion is trivial. Furthermore, G contains no braids and GT naturally seems to introduce them into every situation where it appears. The undertaking therefore framed itself as follows: instead of restricting attention to G, is it possible to extend G by braids, in such a way that it is possible to define a GT-action on a profinite version of the extension, in such a way that the pentagonal relation of G reflects that of GT? The answer turned out to be nearly yes, namely in order to succeed

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it was necessary to use not the group, but the groupoid interpretation of G, in which its elements are considered as morphisms between marked tesselations (this groupoid is known as the Ptolemy groupoid), and to relax the condition of the Ptolemy groupoid stating that the group of morphisms from any marked tesselation to itself is trivial to a condition stating that the group of morphisms from any marked tesselation to itself is isomorphic to a certain ribbon braid group. It is the profinite completion of the ensuing braid-groupoid which admits a GT-action. In §1, we give the definition and presentation of the Ptolemy group, and its interpretation as a groupoid whose objects are marked tesselations of the Poincare disk. In §2, we recall the definitions and important properties of braid and mapping class groups, and their generalizations to ribbon braid and mapping class groups, which will be the braid groups used to extend the Ptolemy groupoid to the Ptolemy-Teichmiiller groupoid. §3 is devoted to the actual construction of the Ptolemy-Teichmiiller groupoid Poo, with a "physical" interpretation of the new, added groups of non-trivial morphisms from a tesselation to itself via braids of ribbons viewed as hanging from the intervals; at the end of the section we define the profinite completion of the groupoid 'Poo by simply taking the profinite completions of each of the local groups, while preserving the set of objects (i.e. marked tesselations) and the basic (Ptolemy) morphisms from one to another. §4 contains the main theorem of the article (theorem 4) explicitly a GT-action on the universal profinite Ptolemy-Teichmiiller groupoid 'PQO ; the role of the two pentagons appears in lemma 5. Finally, in §5 we give a very brief discussion of the relation between the situation considered here and the geometry and arithmetic of genus zero moduli spaces. This article was motivated by the idea of discovering a link between number theory and Penner's universal Ptolemy groupoid, an idea suggested to us by Bob Penner who had himself had conversation with Dennis Sullivan, and immediately seized upon by us because of the distant echo of the pentagonal defining relation of GT which could be heard (upon listening carefully) when considering Penner's sequence of ten moves giving a fundamental defining relation of the Ptolemy group. We recall with great pleasure the enthusiasm of our early discussions with him about the possible links between those relations; later communal discussions with Dennis Sullivan and Vlad Sergiescu were both enlightening and stimulating. We warmly thank all three.

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§1. The universal Ptolemy groupoid A groupoid is a category all of whose morphisms are isomorphisms. We begin by giving some basic definitions leading to the definition of the universal Ptolemy groupoid from [PI, 4.1] (see also [P2]). Identify the Poincare upper half-plane with the Poincare disk via the transformation (z — i)/(z + i). Traditionally, points on the Poincare disk are labeled by the corresponding upper half-plane, so that for instance the points - 1 , —z, 1 and i on the unit circle in C are labeled 0, 1, oo and —1, whereas the central point 0 G C is labeled i. In particular, PXR is wrapped once around the boundary of the disk, the rational numbers of course lie densely in it. We will be particularly concerned with these rational numbers. Let a marked tesselation be a maximal (i.e. triangulating) tesselation of the Poincare disk such that its vertices lie on the set of rational numbers on the boundary, equipped with a directed oriented edge. The standard marked tesselation is the dyadic tesselation T* with the marked edge from 0 to oo:

Figure 1. The standard dyadic tesselation T* with its oriented edge The elementary move on the oriented edge of a tesselation changes it from one diagonal of the unique quadrilateral containing it to the other by turning it counterclockwise; it is of order 4.

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Figure 2. The elementary move on the oriented edge An arrow-moving move on a marked tesselation is an operation on the tesselation which moves the oriented edge to another edge without changing the tesselation itself.

Figure 3. An arrow-moving move Definition: The universal Ptolemy groupoid is the groupoid defined as follows: Objects: The marked tesselations; Morphisms: Finite sequences (or chains) of elementary moves on the oriented edge and arrow-moving moves; Relations: The only morphism from an object to itself is the trivial one. Remark: The condition that the local groups of morphisms (i.e. groups of morphisms from an object to itself) are trivial implies that if T\ and T2 are two marked tesselations, then there is a unique morphism in the groupoid from Ti to T2.

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On any marked tesselation T, let a denote the elementary move on its oriented edge, as shown for the standard marked tesselation in Figure 2. Let the triangle to the "left" of the oriented edge of T denote the triangle lying to our left if we imagine ourselves to be lying face down on the tesselation along the oriented edge, with our head in the direction indicated by the arrow, and let (3 denote the move which sends the arrow counterclockwise to the next edge of the triangle to the left of the oriented edge, as in Figure 3. As noted by Penner, the universal Ptolemy groupoid can be given a group structure, simply because if we write a chain of a's and /?'s, it can be considered as a morphism on any given tesselation: at each point in the chain, the tesselation being acted on is the one resulting from application of all the previous moves. In particular, given a starting tesselation, a word in a and p uniquely determines a morphism in the groupoid, and conversely, every morphism in the groupoid can be given as a starting tesselation and a word in a and /?. Moreover, clearly, if a word in a and /? gives the trivial morphism from some tesselation to itself, then the same word will give the trivial morphisms from every tesselation in the Ptolemy groupoid to itself. We want to find exactly which words these are, namely to determine the relations in the group generated by a and (3 induced by the condition that the group of local morphisms of a tesselation is trivial. In other words, we need to determine the chains of a's and /3's which bring a marked tesselation to itself. Theorem 1. All words in a and j3 taking a given tesselation to itself are generated by the following words (where square brackets denotes the commutator): a 4 , /33, (a/3)5, [pap, a2 papa2],

[pap,

a20a2papa2(32a2].

Proof. In the contribution to this volume by M. Imbert, it is proved that Penner's group G is isomorphic to Richard Thompson's group. Therefore, it suffices to show that the group, let us call it G, defined by generators a and P and relations as in the statement of the theorem is isomorphic to Thompson's group. Let us give a presentation of Thompson's group which can be found on page 2 of Thompson's unpublished notes [T] (and under a different but recognizable notation, [CFP], lemma 5.2.). It is given by three generators, R, D and c\, and six relations, namely: [R'^^D^RDR'1] = 1, [R-1D,R2DR~2] = 1, a = DClR-lD, RDR^DcxR'1 = D2cxR-2D, Rc\ = (DciR'1)2 and c\ — 1. We define a homomorphism from Thompson's group to G by setting (R) = a 2 /? 2 , <j>{D) = a3p and (ci) = /?. We

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need to check first that 0 is really a homomorphism, and second that it is invertible, so an isomorphism. To see that it is a homomorphism it suffices to compute the images of both sides of the six defining relations by (j). All are easily seen to hold in G. Indeed, the first two relations are the analogous commutator relations in G, and c\ = 1 is sent by to f33 = 1. The two sides of the relation cx = Dc1R~1D are sent to (3 and a3(3(3(3a2a3(3 = (3 and the two sides oiRci = (DdR~1)2 are sent to a2 and (a3f3f3f3a2)2 = a2. Finally, rewriting the remaining relation as D2c\R~2DRc~[1D~1RD~1R~1 = 1, we see that the left-hand side is sent by
This shows that cj> is a homomorphism; to show it is an isomorphism, it suffices to define (j)~1(a) = c\D~l and <j)~1(f3) = c\ (indeed, the relation c\ = Dc\R~1D shows that Thompson's group is generated by the two elements c\ and D). A Remark. The generators of this group can be identified with the corresponding moves on marked tesselations in the Ptolemy groupoid. Indeed, the words a4 and (33 clearly bring a marked tesselation back to itself and are therefore trivial; similarly (a/3)5 — 1 corresponds exactly to Penner's trivial sequence of 10 moves (cf. (c) on p. 179 of [PI]; note that (a/3) simultaneously moves the two diagonals of a pentagon whereas Penner moves one at a time). Finally, the two commutation relations in G imply that that elementary moves on the diagonals of two neighboring quadrilaterals commute, and elementary moves on the diagonals of two quadrilaterals separated only by a triangle; it is a remarkable fact that the commutation of all pairs of elementary moves taking place in disjoint quadrilaterals (Penner's relation (d) on p. 179 of [P]) are consequences of the relations in G. §2. Ribbon braids Let us recall the definitions of the Artin ribbon braid groups and the ribbon mapping class groups. First recall the definitions of the usual Artin braid and mapping class groups. The Artin braid group Bn for n > 1 is generated by <7i,..., 2.

There is a natural surjection p : Bn -> Sn for n > 1 which induces a natural surjection p : M(0, n) —> Sn (for n = 4 we have the surjection p :

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B3/Z -> S3). The kernel of p (denoted by Kn in Bn and K(0, n) in M(0, n)) is known as the pure braid group resp. mapping class group. Both Kn and K(0,n) are generated by the elements Xij := o~j-i--ai+iafa^--a^\ for 1 < i < j < n. The center of Bn and of Kn is cyclic generated by the element ujn = £i2#i3#23 • °°xln • • 'Xn-i,n- The mapping class group M(0, n) (resp. the pure mapping class group if (0, n)) is the quotient of Bn (resp. if n ) by the following relations: (i) ujn = 1; (ii) Xiii+ia;i>t+2 • "Xi^x^i • • -x^i-i = 1; for 1 < i < n, where the indices are considered in Z/nZ. The Artin ribbon braid group i?* is a semi-direct product

it is generated by generators <7i,..., crn_! of the £ n factor and f 1,..., tn (all commuting) of the Z n factor, with the "semi-direct" relations given by: (Titia~l = £»+i for 1 < i < n — 1 (Jiti+io~~l = ti for 1 < i < n — 1

{(Ti,tj) — 1 for 1 < i < n - 1, j ^ i, i + 1 This group is visualized like the usual braid groups, except that the strands are replaced by flat ribbons, so that a twist on any one of them is non-trivial. The subgroup of B^ consisting of "flat" braids of the ribbons (i.e. without twists on the ribbons, cf. Figure 4) is canonically isomorphic to Bn, and from now on we identify Bn and its generators Oi with this subgroup of £*.

Figure 4. The flat braid

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We identify Z n with the abelian subgroup of i?* generated by a full (2?r) twist ti on each of the n ribbons (cf. Figure 5).

Figure 5. A full twist on a ribbon This visualization corresponds to the definition of #* as a semi-direct product ZnxBn given above. Since the twists ti commute with pure braids, the pure ribbon braid subgroup if* of £?* is just a direct product Z n x Kn. Let us define the ribbon mapping class group M*(0, n) to be B* modulo the following relations. Firstly, the center of J9* is generated by the element which also generates the center of ifn, namely ujn = X12X13X23 • • 'X\n • • *# n _i ?n , together with the subgroup Z n . The first relation we quotient out by is

Now, instead of using the usual sphere relations by which we quotient Bn to obtain M(0, n), we use the ribbon sphere relations: {\\') Xiyi+iXij+2 ' ' ' Xi,n%l,i ' ' ' %i,i-l =

tj.

The quotient of B* by the relations in (i') and (ii;) is the ribbon mapping class group M*(0, n). The surjection of Bn onto Sn extends to £?* by sending the subgroup Z n to 1, and the kernel of this surjection is the pure ribbon braid group K^. The surjection passes to M*(0, n) and its kernel in this group is denoted by i
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Let us give some admirable properties of the ribbon groups. (1) It is well-known that removing any strand gives a surjection from Kn onto if n _i, which induces a surjection from if (0, n) onto if (0, n— 1). There are analogous natural surjections from if* into K*_x and from K *(0, n) into if*(0, n — 1) obtained by removing one ribbon. Removing several ribbons thus gives surjections from K*(0,n) onto if*(0,ra) for m < n. (2) The braid obtained by holding two adjacent ribbons i and i + 1 firmly by their bottom ends and twisting them one full turn is equal to Uti+iXij+i. We denote this ribbon braid by U^+i. It is the same as the full twist on the single "wide" ribbon obtained by sewing the two adjacent ribbons together. The expression for the simultaneous full twist of several adjacent ribbons can easily be deduced from this one by induction. (3) There are natural injections Km into Kn for m < n given by dividing up the n strands into m adjacent packets (each of which can consist of one or more strands) called A±,... ,Am; the subgroup of Kn generated by the "flat" braids XA^AJ (as in Figure 4, considering each packet as a ribbon) is isomorphic to Km. Analogously, there are natural injections of K^ into if* for any division of the n ribbons into m adjacent packets. Each packet, considered as adjacent ribbons sewn together, forms a "wide ribbon", and the group if^ of braids on these wide ribbons is naturally a subgroup of Now, there is no such natural injection for the pure mapping class groups if (0, n), because the twist on a packet of strands is non-trivial whereas the twist on a single strand is trivial. This problem is eliminated for the ribbon mapping groups where the ribbons and the wide ribbons behave in the same way with respect to twists. Therefore we have such injections for the pure ribbon mapping class groups if *(0, m) ^^ if *(0, n); this point represents the major advantage of the use of ribbon braid groups with respect to ordinary braid groups.

§3. The universal Ptolemy-Teichmiiller groupoid The universal Ptolemy-Teichmiiller groupoid T^oo is a generalization of the universal Ptolemy groupoid in the sense that we add morphisms from a given marked tesselation to itself. The objects of Poo are those of the universal Ptolemy groupoid, namely marked braid tesselations; what we do

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here is to relax the condition stating that the local groups are all trivial to a condition defining the local groups as certain ribbon braid groups. To explain exactly what is going on, we visualize a marked tesselation T a little differently; we assume that there is a ribbon hanging from the interval on the circle delimited by each edge of the tesselation. To be precise, each edge of the tesselation actually divides the circle into two intervals, and we want to choose only one of them; we choose to hang the ribbon from the interval lying entirely on one side of the oriented edge. This makes sense for every edge except the oriented one, to which we associate the interval lying to the left of it in the sense explained earlier. Note that since the ribbons are determined by the edges of T, assuming their presence does not add anything to T; the point of adding them is that the nontrivial morphisms from T to itself which we are going to introduce correspond exactly to braiding them. Before continuing, we note that if one considers the infinite trivalent tree dual to the tesselation, then it comes to the same thing to attach a strand to each of its "ends" (the rationals) and consider the set of strands in a given interval as forming a ribbon, and this in turn is equivalent to attaching a strand to each vertex of the tree (uniquely associated to a rational). This idea, due to Greenberg and Sergiescu (cf. [GS]) was one of the starting points of this article. Consider thus from now on each marked tesselation T to come equipped with its ribbons. Each ribbon is automatically associated to an interval on the circle (delimited by an edge of T, a fortiori by two numbers in PXQ). Two ribbons of T are said to be disjoint if their intervals are disjoint except for at most one common endpoint. They are said to be neighbors if their associated intervals are disjoint except for exactly one common endpoint. Two ribbons of T are said to be adjacent if their intervals are delimited by two sides of a triangle of the tesselation; thus, adjacent ribbons are of course neighbors and neighbors are disjoint, but the converses are not necessarily true. Two neighboring ribbons of a given marked tesselation can always be made into adjacent ribbons of another tesselation which can be obtained from the first by a finite number of elementary moves, successively reducing to zero the (finite) number of edges coming out of the common endpoint of the two ribbons and lying inside the smallest polygon of the tesselation having as two of its edges those associated to the ribbons. On the left-hand side of Figure 6, we show two neighboring ribbons; the smallest polygon of the tesselation containing having their corresponding edges as edges is a

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quadrilateral and there is just one edge coming out of the common endpoint of the two ribbons and lying inside it (namely, its diagonal). Thus, after the elementary move on that diagonal, the two ribbons become adjacent (right-hand side), associated to two sides of the triangle A.

Figure 6. Adjacent and neighboring ribbons of a tesselation Definition: Let A and B be disjoint ribbons of a given marked tesselation T. Let t\ denote the full twist on A, oriented as in Figure 5, and let xTAB denote the flat braid of A and B shown in Figure 7, where the ribbon on the left-hand side passes in front of the right-hand one (whether the observer stands inside or outside the tesselation).

Figure 7. The braid xTAB

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The local group KT at a marked tesselation T is a group of morphisms from T to itself, essentially given by braiding the ribbons associated to intervals of T; it is defined as follows. Definition. Let KT be generated by the flat braids xTAB for all pairs of disjoint ribbons A and B of T and by the full twists tTA on each of these ribbons. Define the set of relations in the group KT to be the set of relations coming from the finite polygons of T containing the oriented edge, as follows. Let S be such a polygon, say with n sides, and let A\,..., An be the n ribbons associated to the intervals determined by the sides; they are pairwise disjoint (which would not be the case if the polygon S did not contain the oriented edge of T and thus lay entirely on one side of it). Let KT(S) denote the subgroup of KT generated by the twists t^. and the flat braids xTA. A.. Then the relations of KT are generated by all the relations induced by the assertion: For every S, the group KT(S) is isomorphic to the pure ribbon mapping class group K*(0,n). Definition: The universal Ptolemy-Teichmuller groupoid Voo is defined as follows: Objects:

Marked tesselations.

Morphisms: They are of two types: firstly, those of the universal Ptolemy groupoid, which act on marked tesselations as usual, and secondly, the groups Hom(T, T) ~ KT of morphisms from each marked tesselation to itself. Relations: The full set of relations in the universal Ptolemy-Teichmuller groupoid VQO is given by: (i) those of the universal Ptolemy groupoid (any sequence of elementary moves leading from a tesselation to itself is equal to 1); (ii) those of the local ribbon braid groups; (iii) commutativity of these two types of morphisms as in equation (1) below. The universal Ptolemy-Teichmuller groupoid contains the universal Ptolemy groupoid as a subgroupoid, because of (i). Let us explain (ii) and (iii) further. Firstly, from now on we use the term interval to denote an interval of the circle delimited by two rationals, and an interval ofT or equivalently,

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a ribbon of T to denote the interval delimited by an edge of the tesselation T, as before (the one on the opposite side from the oriented edge of T). Not every interval of the circle is an interval of T, of course, but every interval is the union of a finite number of intervals of T; we call such an interval a wide interval or a wide ribbon of T (obtained by sewing together a finite number of neighboring ribbons of T). Thus every interval of the circle is associated to a ribbon or a wide ribbon of T. Let A and B denote two disjoint intervals (recall that "disjoint" intervals may have one endpoint in common), equipped with ribbons. Let the braid X AB denote the usual flat braid (as in Figure 7); this twist can be applied to ribbons corresponding to any two disjoint intervals of the circle, without needing to refer to a specific tesselation. However, fixing a tesselation T, we see that the ribbons associated to the intervals A and B are either ribbons or wide ribbons of T, which implies that the braid XAB actually lies in KT for all T. We write xTAB when we want to consider the braid XAB Q>S an element of KT. Recall that given marked tesselations T' and T, there is a unique morphism 7 in the universal Ptolemy groupoid from V to T. This gives rise to a canonical isomorphism between KT and KT via KT = j~1KT'y. For all pairs of disjoint intervals A and B of the circle, this isomorphism has the property that *

=7~1*7

(1)

in the universal Ptolemy-Teichmiiller groupoid. This is what is meant by the commutation of the morphisms of the universal Ptolemy groupoid with braids in (iii) above. Proposition 2. Let T be a marked tesselation. (i) A set of generators for the group KT is given by the set of braids xTAB for all pairs of ribbons A and B corresponding to disjoint intervals of the circle, and the twists tTA on the wide ribbons of T corresponding to all intervals of the circle. The set of relations associated to this set of generators is independent of T. (ii) A smaller set of generators for KT is given by the twists tTA and xTAB where A and B are disjoint ribbons of T. A set of relations for KT associated to this set of generators is given by the relations between the generators of each KT(S) for finite polygons S of T. (iii) Another set of generators for KT is given by the set of twists t^ on

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all ribbons ofT and xTAB where A and B are pairs of adjacent or neighboring wide ribbons ofT. Indeed, if S a finite polygon ofT containing the oriented edge and the ribbons of S are the ribbons of T associated to the edges of S, then each subgroup KT(S) is generated by the twists on ribbons of S and the xTAB where A and B are adjacent or neighboring wide ribbons of S.

Proof, (i) and (ii) are immediate consequences of the definition of KT. For (iii), we start by showing the statement for KT{S) C KT. By definition, this subgroup is isomorphic to K*(0,n) where n is the number of edges of the polygon S. A set of generators for the pure Artin braid group Kn is given by the elements Xi...jj+i...k for all 1 < i < j < k < n; this braid denotes the flat braid of the "neighboring packets" of strands numbered i • • • j and j - f l • • • fc, which can be considered as ribbons; it looks like the one in Figure 4, with one or several strands in place of the ribbons. To see that this set really generates, it suffices to write each of the usual generators Xij of Kn in terms of these, which can be done via the formula x

ij —

x

i---j-l,jxi+l...j-lj

(draw the picture!) If j = i + 1, the usual x^ is itself a twist of neighboring packets, which consist of one strand each. The flat braids of neighboring packets Xi...jj+i...k also generate the quotient if (0, n) of Kn, so adding in the full twists on ribbons, we have a set of generators for if*(0,n). Since this group is isomorphic to KT(S), this proves the statement of (iii) for the groups KT(S). It follows immediately for KT since the group KT is generated by the subgroups KT(S) as S runs over all the finite polygons of T containing the oriented edge of T. A Let us now describe the "profinite completion" of the universal PtolemyTeichmuller groupoid VOQ. We need the following: Lemma 3. (i) If S and R are finite polygons containing the oriented edge of a given marked tesselation T\ and S lies inside R, then the subgroup KT(S) C KT is contained in KT(R). (ii) For n > 4, let Sn denote the 2n-gon in the standard marked dyadic tesselation T* obtained from S4, the basic quadrilateral containing the oriented edge Ooo, by successively dividing every edge into two. Then

KT' =

\jKT'(Sn), n>2

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where this union of groups is given by the natural inclusion of KT (Sn) into KT (£ n +i) induced by the inclusion of Sn in 5n_j-i, as in (i). (Hi) Let KT*(Sn) denote the profinite completion of KT*(Sn) forn > 2. Then the inclusion of Sn into S n +i induces a natural inclusion of KT (Sn) intoKT\Sn+1). Proof. Part (i) follows from the existence of injections if*(0, m) —> i^*(0, n) for m < n, sending ribbons in i^*(0, m) to wide ribbons in if*(0,n), cf. property (3) in §2. Part (ii) is a corollary of this, since every polygon of T* lies inside Sn for some sufficiently large n. Finally, (iii) is a consequence of the fact that like all braid and mapping class groups, the KT* (Sn) inject into their profinite completions. A Let us define the profinite local group at T* by

n>2

For any pair of disjoint intervals A, B of the circle, there exists n such that xT^B lies in iir T *(5 n ), since every rational is a vertex of some Sn. Since KT* (Sn) injects into its profinite completion, the braid xT^B lies in KT (Sn). The group KT is topologically described by the same set of generators and relations as KT . We define the profinite local group at any marked tesselation T to be the one obtained from KT* as in equation (1), i.e. by conjugating by the unique morphism 7 in the universal Ptolemy groupoid which takes T* to T. Definition: Let the profinite completion V^ of Poo be the groupoid defined as follows: Objects:

Marked tesselations T.

Morphisms: All the morphisms of the universal Ptolemy groupoid, together with the local groups Hom(T, T) = Kj, at each T. Relations: (i) those of the universal Ptolemy groupoid; (ii) those of the local profinite braid groups; (iii) commutativity of these two types of morphisms as in equation (1).

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§4. GT and the automorphism group of VOQ Definition: The automorphism group Aut°('Poo) of the completed Ptolemy groupoid Poo is defined to be the set of automorphisms of the groupoid P ^ which act trivially on the set of objects. The goal of this article is to show that V^, considered as a completion of the universal Ptolemy groupoid, has the property that the GrothendieckTeichmuller groupoid lies in its automoprhism group Aut°('Poo) (cf. §0). To prove this, we begin by recalling the definition of GT (cf. the survey [S] for references and details). Definition: Let GT be the monoid of pairs (A, / ) G Z* x F^ satisfying the three following relations, the first two of which take place in the profinite completion F2 of the free group on two generators F2, and the third in the profinite completion K(Q,5) of the pure mapping class group: (I) /(y, *)/(*, y) = l; (II) f(z,x)zmf(y,z)ymf(x,y)xm

= 1, where m = ±(A- 1) and z = {xy)'1]

(III) /(X34,X45)/(x5i,3;i2)/(a;23)^34)/(^45^5l)/(^12,^23) = 1.

Under a suitable multiplication law, this set forms a monoid. The group GT is defined to be the group of invertible elements of the monoid GT. Drinfel'd and Ihara showed that the group GT contains the absolute Galois group Gal(Q/Q) as a subgroup in a natural way (again, cf. [S] for all relevant references). In particular, whenever GT acts on an object, this object becomes equipped with a Galois action, indicating a - sometimes quite unexpected - link with number theory, which was one of the main motivations behind this article. Theorem 4. There is an injection GT <-•» Aut°('Poo ). Proof. Let F = (A, / ) e GT. Then we let F act trivially on the set of objects of VOQ. The proof is outlined as follows: first we define the action of F to be trivial on arrow-moving morphisms, next we give its definition on the morphisms of the universal Ptolemy groupoid (considered as a subgroupoid of T^oo) and prove in proposition 5 that the relations of the universal Ptolemy groupoid are respected, andfinallywe define the action on a set of generators of the profinite local group KT - using lemma 5 to show that the action on the generators is well-defined - and prove in lemma 6 that the action extends to an automorphism of KT'. This takes care of two of the three types of morphisms in VOQ\ to conclude, we show that the commutation

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relations of equation (1) are respected. Let us proceed to the definition of the action of F on the morphisms of Arrow-moving morphisms. F acts trivially on these. Elementary morphisms. The oriented edge of a marked tesselation T determines two pairs of adjacent edges forming a quadrilateral called the basic quadrilateral of T; we call the ribbons hanging from the intervals delimited by these four edges XT, YT, %T and WT respectively, going around counterclockwise from the point of the arrow. Let ar denote the elementary move on the oriented edge of T. We set F(aT) = aT • f(xTXTYT,xlTZT). The profinite braid /(^x T y T ' x ?T^T) *s a in KT\ so F((XT) is a morphism of Poo.

mor

(2)

P m s m from T to itself, lying

Figure 8. The basic quadrilateral of a marked tesselation Lemma 5. This action of GT respects all the relations of the universal Ptolemy groupoid. Remark. The lemma can be restated by saying that every element of GT is a groupoid-isorphism from the universal Ptolemy subgroupoid of Poo to its image. This shows that GT respects the first of the three types of relation in Poo.

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Proof. The point is to check that if F = (A, / ) is an element of GT, then its action on any two finite sequences of elementary moves leading from T to T' is the same, since in the universal Ptolemy groupoid two such sequences give the same morphism. It is equivalent to check that F respects the relations in the groupoid, and these relations were given explicitly in theorem 1. So we simply need to compute the action of F on the five relations in a and /? given in the statement of theorem 1. Here we index the moves according to the tesselation they are acting on, so that the group can be recovered from the groupoid by dropping all indices. The relation /33 = 1 is trivially satisfied since F fixes /?. It is also clear that F respects the two commutation relations since they involve braids on disjoint ribbons (or at worst, one braid is made of ribbons all of which are contained in a single ribbon of the other braid) and such braids commute in the local braid groups. As for the pentagon relation (a/?)5 = 1, or equivan d five repeated alency (/?a)5 = 1, we have F(/3a) = (3af{xxY^Yz)i applications of this map, together with the use of equation (1) to push all the factors of /3a to the left, leave us with exactly the famous pentagon relation (III) defining GT, equal to 1. Let us check the remaining relation, a4 = 1. To start with, fix a marked tesselation To, so that OLT0 and /3T0 are the moves shown in figures 2 and 3. Then by equation (2), F(aTo) = aTo f (^^YT^ xYrQ ZTQ) - L e t Tu T2 and T3 denote the tesselations obtained from TQ via ay 0 , a^ &T0 and (XT2 OLTX &T0 (i.e. a, a2 and a 3 in the group), and write cti — a^., so that

asa2aiao = 1. Note that for the four tesselations T;, we have

where Xi, Yi, Zi and Wi are the ribbons attached to the four intervals of the each tesselation Ti shown in figure 8. Applying F to this relation, we obtain F(a4) = F(a3a2aia0)

= a3a2aia0

=

= a4 = 1,

since / satisfies f(x,y) = f(y,x)~l of the relation a4 — 1.

by relation (I) of GT. This takes care A

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Braids: Let us define the action of GT on the local groups KT. We begin by defining an action on a set of generators of KT. By (iii) of proposition 2, a set of (topological) generators of KT is given by the braids xTAB where A and B are neighboring wide ribbons of T, i.e. (finite) unions of neighboring ribbons corresponding to neighboring edges of a finite polygon of T containing the oriented edge. An example is shown in figure 9.

Figure 9. Neighboring intervals corresponding to wide ribbons of T To define the action of F on all the xTAB for neighboring wide ribbons A and B of T, it suffices to define it only on the xrAB for adjacent (possibly wide) ribbons A and B (recall that adjacent ribbons are ribbons attached to intervals corresponding to two edges of a triangle of T) for all tesselations T; we can then extend it to pairs of neighboring wide ribbons by equation (1) and the fact that we know the action of F on morphisms of the universal Ptolemy groupoid. This works as follows. Firstly, if A and B are adjacent (possibly wide) ribbons of T we set V AH/

\

AD

)

'

\

)

If A and B are neighboring (possibly wide) ribbons of T, i.e. (unions of) intervals corresponding to edges of a finite polygon S of T, then we change T into another tesselation T" such that the intervals A and B are two edges of a triangle of T", via a finite number of elementary moves on T, all taking place inside S. By writing them down explicitly and using (2) and the action of GT on elementary moves, we can compute the explicit expression

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for F(x\B) as follows. Choose a finite sequence of marked tesselations T* = T 0 , . . . , T r such that (1) for i > 0, Ti is obtained from T^_i by one elementary morphism gi on some edge lying inside 5 (not an edge of 5); (2) each Ti contains the polygon 5 and is identical to T* outside of 5; (3) the ribbons corresponding to the intervals A and B are adjacent ribbons of T r , i.e. the intervals A and B are two sides of a triangle of Tr. Let 7 = gr o gr_i o • • • #i; then no matter what choices we make for T i , . . . , Tr and # i , . . . , # r , 7 is the unique morphism of the universal Ptolemy groupoid taking T* to Tr. By equation (1), we have ^f~lxAB^ = XAB. By repeated applications of equations (1) and (2), we find an element 77 G K*(0,s) C K^* such that ^(7) = 777; the fact that 77 is well-defined is a consequence of lemma 5. Thus, the action of F on X\B when A and B are neighboring wide ribbons is given by

F(xTAB) = T T V ^ B ) ^ = T ^ L O V

(4)

#; This action of GT on the generators of VQQ extends to a groupoid automorphism of V^. We must check that all the relations of the groupoid are respected. Lemma 6. The action defined above ofGT on the generators xTAB of K^ for all pairs of adjacent or neighboring clumps A and B of T* determines an automorphism of Proof. Let F G GT. Then the action of F on the generators of each of the groups K*(0, Sn) for n > 2 extends to an automorphism. Indeed, it is known (cf. [PS], chapter II) that GT is an automorphism group of the pure mapping class group i^(0,2 n ) in many ways, corresponding to the trivalent trees with 2 n edges; the action we consider here corresponds to the trivalent tree dual to the polygon Sn. Now, we have the exact sequence

and it is easily seen that the GT-action on K(0,2n) extends to an automorphism of K*(Q, Sn) simply by letting F = (A, / ) eGT act on each twist tTA by sending it to {t^)x. Now let us show that the automorphisms of each lf*(0, Sn) given by F eGT respect the natural inclusions ^ ( O , ^ ) <->• K*(0,Sn+1). Recall

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that S n +i is obtained from Sn by subdividing each interval of Sn into two. If A and B are neighboring wide ribbons of T, then xTAB lies in K*(0, Sn) if and only if A and B are actually supported on S n , i.e. correspond to intervals delimited by a finite number of neighboring edges of Sn. Supposing this is the case, then of course xTAB is also supported on Sn+i, so xTAB also lies in K*(0,Sn+i), as it should since K*(0,Sn) injects into if*(0, S n +i). Furthermore, in order to compute F(x'AB), we need to use a finite series of elementary morphisms as explained in the definition of the GT-action on braids, and they consist of moves on edges lying in 5 n , and are therefore the same whether xTAB is considered as lying in if*(0, Sn) or jfrT*(O, Sn+i), so that the expression of F(x'AB) is not dependent on n, i.e. F respects the injection £*(0,S n ) -> K*(0,Sn+1). Q Lemma 7. For A and B disjoint intervals of the circle and T and T' different marked tesselations, the commutativity relations xTAB — 'Y~1X'AB'Y of equation (1) are respected by the action of GT. Proof. Recall that 7 is a finite chain of morphisms in the universal Ptolemy groupoid taking T' to T. In the case where A and B are actually adjacent ribbons for T, the lemma follows immediately from the definition of the action of F on the braids xTAB. If A and B are not adjacent ribbons for T, it suffices to take a third tesselation T" such that they are adjacent ribbons for T" and then again use the definition of F on the braids xTAB and xTAB, by commuting them to Tn via an element of the universal Ptolemy groupoid.

0 Lemmas 5, 6 and 7 show that the action of GT respects all defining relations of the universal Ptolemy-Teichmiiller groupoid, and this concludes the proof of theorem 4. A §5. Relations with the ordered Teichmiiller groupoids Let us very briefly sketch the relationship between the universal PtolemyTeichmuller groupoid and the fundamental Teichmiiller groupoids of genus zero moduli space. Let A^o,n denote the moduli space of Riemann spheres with n ordered marked points. The Teichmiiller groupoids are the fundamental groupoids 7Ti(.Mo,n; #n) of the moduli spaces A1o,n for n > 4 of genus zero Riemann surfaces with n ordered marked points, on the set Bn of base points near infinity of maximal degeneration. The use of these groupoids

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was suggested in [D], page 847, and their structure was investigated in detail in [PS], chapters 1.2 and II. The set Bn is essentially described by isotopy classes of numbered trivalent trees with n leaves. Let us define ordered Teichmuller groupoids to be the fundamental groupoids of the moduli spaces Mo,n on a certain subset Cn of the base point set 6 n , so that the ordered Teichmiiller groupoids are subgroupoids of the full Teichmuller groupoids. We define the base point sets Cn of the ordered Teichmuller groupoids to be the set of base points near infinity in Moyn corresponding to trivalent trees whose n leaves are numbered in cyclic order 1,..., n. The set of associativity moves acts transitively on such trees, so that the paths of the ordered Teichmuller groupoids are of two types: the braids (local groups), i.e. the fundamental groups of M 0 ,n based at each base point, which are all isomorphic to if(0,n), and associativity moves going from one base point to another. The universal Ptolemy-Teichmuller groupoid covers all the ordered Teichmiiller groupoids for n > 4 in the sense that these groupoids naturally occur as quotients of subgroupoids in many ways. Indeed, for n > 4, choose a n-sided polygon S in any given tesselation T, and consider the set of elementary paths on T which act only on edges inside the polygon. In other words, consider the finite set of tesselations X" differing from T only inside the polygon S. Now consider the set of marked tesselations obtained from these by marking any chosen edge of T not in the interior of S, and the same edge on the other tesselations differing from T only inside S. This gives a subgroupoid of 'Poo on a finite number of base points. Now we quotient the local group at each tesselation by suppressing all the ribbons except those attached to intervals delimited by edges of the polygon S. The quotient of Kj> obtained in this way is exactly the pure mapping class group KT(S) on these ribbons, isomorphic to K*(0, n); thus we obtain the ordered Teichmuller groupoid as a quotient of Poo. It is shown in [SI] that there is a GT-action on the proflnite completion of the fundamental groupoid 7Ti(A^o,n;^n); this action fixes the objects of the groupoid, i.e. the elements of B n , so it restricts to an action of 7ti(Mo,n;Cn). The relation with the main theorem of this article is that our GT-action on 'PQQ passes to the quotient described here, and gives exactly the usual one on 7Ti(Mo,n;Cn).

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References. [CFP] J.W. Cannon, W.J. Floyd and W.R. Parry, Notes on Richard Thompson's groups, preprint (University of Minnesota, 1994), to appear in Enseignement Math. [D]

V.G. Drinfel'd, On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(Q/Q), Leningrad Math. J. 2 (1991), 829860.

[GS] P. Greenberg and V. Sergiescu, An acyclic extension of the braid group, Comm. Math. Helv. 66 (1991), 109-138. [IS]

M. Imbert, Sur l'isomorphisme du groupe de Richard Thompson avec le groupe de Ptolemee, this volume.

[MS] G. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory, Commun. Math. Phys. 123 (1989), 177-254. [PI]

R.C. Penner, Universal Constructions in Teichmiiller Theory, Adv. in Math. 98 (1993), 143-215.

[P2]

R.C. Penner, The universal Ptolemy group and its completions, this volume.

[PS]

Triangulations, courbes arithmetiques et theories des champs, ed. L. Schneps, issue to appear of Panoramas et Syntheses, Publ. SMF, 1997.

[S]

L. Schneps, On GT, a survey, Geometric Galois Actions, volume I.

[SI] [T]

, On the genus zero Teichmiiller tower, preprint. R. Thompson, unpublished handwritten notes.

URA 762 du CNRS, Ecole Normale Superieure, 45 rue d'Ulm, 75005 Paris UMR 741 du CNRS, Laboratoire de Mathematiques, Faculte des Sciences de Besangon, 25030 Besangon Cedex

Errata for Tame and stratified objects* Bernard Teissier I wish to thank Selma Kuhlmann for bringing to my attention some errors and omissions in the text. Page Page Page Page Page Page Page

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This article appeared in volume I of Geometric Galois Actions.