Groups, Trees and Projective Modules

Lecture Notes in lVlathematics Edited by A. Doid and B. Eckmann

790 Warren Dicks

Groups, Trees and Projective Modules

Springer-Verlag Berlin Heidelberg New York 1980

Author Warren Dicks Department of Mathematics Bedford College London NWt 4NS England

AMS Su bject Classifications (1980): 16 A 50, 16 A ?2, 20 E 06, 20 J 05 ISBN 3-540-099?4-3 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-09974-3 Springer-Verlag NewYork Heidelberg Berlin Library of Congress Cataloging in PublicationData.Dicks, Warren, 1947- Groups, trees, and projectivemodules.(Lecture notes in mathematics;790) Bibliography:p. Includes indexes.1. Associativerings. 2. Groups, Theoryof. 3. Trees (Graph theory)4. Projective modules (Algebra) I. Title. II. Series: Lectures notes in mathematics(Berlin); ?90. O.A3.L28 no. 790. [QA251.5], 510s. [512'.4]. 80-13138 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 214113140-543210

To the memory of my mother

PREFACE

For 1978/9 the Ring Theory Study Group at Bedford College rather naively set out to learn what had been done in the preceding decade on groups of cohomological dimension One. attractive subject,

This is a particularly

that has witnessed substantial success,

essentially beginning in 1968 with results of Serre, Stallings and Swan,

later receiving impetus from the introduction of the concept

of the fundamental group of a connected graph of groups by Bass and Serre,

and recently culminating in Dunwoody's contribution which

completed the characterization.

Without going into definitions,

one can state the result simply enough: (associative, with 1) and group group ring

RIG]

is right

G,

For any nonzero ring

R

the augmentation ideal of the

R[G]-projective if and only if

G

is

the fundamental group of a graph of finite groups having order invertible in

R.

These notes,

a (completely) revised version of those prepared

for the Study Group,

collect together material from several

sources to present a self-contained proof of this fact,

assuming

at the outset only the most elementary knowledge - free groups, projective modules, etc.

By making the role of derivations even

more central to the subject than ever before, simplify some of the existing proofs,

we were able to

and in the process obtain

a more general "relativized" version of Dunwoody's result, IV.2.10.

~

cf

amusing outcome of this approach is that we here have

a proof of one of the major results in the theory of cohomology of groups that nowhere mentions cohomology - which should make this account palatable to hard-line ring theorists.

(Group theorists

VI

PREFACE will notice we have not touched upon the fascinating subject of ends of groups~ cf

usually one of the cornerstones of this topic,

Cohen [72];

happily,

an up-to-date outline of the subject

of ends is available in the recently published lecture notes of Scott-Wall [79].) There are four chapters. sections, trees

Chapter I covers,

in the first six

the basics of the Bass-Serre theory of groups acting on

(using derivations to prove the key theorem, 1.5.3),

and

then in I.§8, I.§9 gives an abstract treatment of Dunwoody's results on groups acting on partially ordered sets with involution. Chapter II gives the standard classical applications of the BassSerre theory~

including a proof of Higgins'

generalization of the

Grushko-Neumann theorem (based on a proof by I.M.Chiswell). Chapter III presents the Dunwoody-Stallings decomposition of a group arising from a derivation to a projective module, Dunwoody's accessibility criteria.

Finally,

and gives

in Chapter IV,

the

groups of cohomological dimension one are introduced and characterized;

the final section describes the basic consequences

for finite extensions of free groups. A reader interested mainly in the projectivity results of IV.§2 can pursue the following course:Chapter I:§§i-6,§8,§9; Chapter II:3.1,3.3,3.5;

Chapter III:I.I,I.2,§2~§3,4.1-4.8,4.11,

Chapter IV:§I,§2. Since the subject is quite young, extent still tentative,

and the notation to some

we have felt at liberty to introduce new

terminology and notation wherever it suited our needs, satisfied our category-theoretic prejudices.

or

At these points,

we

have made an effort to indicate the notations used by other authors.

Vll PREFACE Through ignorance, of historical since,

remarks,

we have been unable to give much in the way and those we have given may be inaccurate,

as both Cohen and Scott have remarked,

attribute,

with any precision,

in the literature

it is difficult to

results which existed implicitly

before being made explicit.

The computer microfilm drawings,

pp

13, 25,

were produced by

the CDC 7600 at the University of London Computer Centre, their copyrighted

software package DIMFILM.

and Phil Taylor for their helpful technical

using

! thank Chris Cookson advice in using this

package. I thank all the participants

of the Study Group for their kind

indulgence

in this project,

and especially Yuri Bahturin and

Stephenson

for relieving me (and the audience)

Bill

by giving many of

the seminars. I gratefully acknowledge encouragement)

much useful background

from the experts at Queen Mary College,

Chiswell and D.E. Cohen for Chapters Bedford College London January 1980

material

(and

I.M.

l-II and III-IV respectively. Warren Dicks

NOTATION

The

following

notation

will

AND

CONVENTIONS

be used:

for the empty

set;

for the ring of integers;

£

for the field

of r a t i o n a l

for the

of complex

field

for the set of elements

A - B

numbers; numbers;

in

A

not

in

B;

IAJ

for the

BA

for the set of all functions from A to B, the elements thought of as A-tuples with entries chosen from B;

A × B,

cardinal

for the C a r t e s i a n

IIB

of

A;

product;

~eA ~ A v B,

for the disjoint

V B

union

of sets;

o~eA c~ A @ B,

for the

@ B

direct

sum of modules.

~eA a Functions their

are usually,

conventions

outside

Chapter

end of each

primes

follows

lemmas,

consecutively 4.2

is indicated

to the b i b l i o g r a p h y

on the right

by

publications

remarks

in each section, in section

to as 1.4.3

of

1.4

and

thus (and

and 1.4.2).

The

D.

are by author's

of the y e a r of publication,

to distinguish

corollaries,

DEFINITION

I they are r e f e r r e d

subsection

References two digits

propositions,

are n u m b e r e d

CONVENTION

year.

written

arguments.

All theorems,

4.3

but not always,

thus

Serre

name and the last [77],

by the same author

with

in the same

CONTENTS

CHAPTER 1.1 1.2 1.3 1,4 1,5

1.6 1.7 1.8 1.9

II:

The trivial Basic results The faithful Coproducts

ON

GRAPHS

...... 1 @ @

FUNDAMENTAL case

III:

.

case . . .

IV:

4 7 10

15 21

24 27 31

GROUPS

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

free groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

DECOMPOSITIONS

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

Decomposing a group . . . . . . . . . . . . . . Cuts . . . . . . . . . . . . . . . . . . . . . . Oecomposition theorems . . . . . . . . . . . . The relationship with derivations . . . . . Accessibility . . . . . . . . . . . . . . . .

CHAPTER IV.1 IV.2 IV.3

ACTING

The structure theorem , An e x a m p l e : S L 2 ( ~ ) .. Fixed points . . . . . . . T r e e s and p a r t i a l o r d e r s

CHAPTER III.1 III.2 III.3 III.4 III.5

GROUPS

Graphs . . . . . . . . . . . . . . . . . . . . Graph morphisms and coverings . . . . . . Group actions . . . . . . . . . . . . . . Graphs of groups . . . . A tree . . . . . . . . .

CHAPTER II.1 II.2 II.3 II.4

I:

COHOMOLOGICAL

DIMENSION

ONE

. . . . . . . . . .

. . . . . . . . . . .

35 35 38 43 49

55 55 62 68 74 82

101

........

Projective augmentation modules . . . . . . . . . . . . P a i r s of g r o u p s . . . . . . . . . . . . . . . . . . . . F i n i t e e x t e n s i o n s of free g r o u p s . . . . . . . . . . . .

102 107 118

BIBLIOGRAPHY

121

SUBJECT

SYMBOL

INDEX

INDEX

AND

AUTHOR

INDEX

.

.

.

.

.

.

.

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

.

.

.

.

.

.

.

.

.

.

.

.

.

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

125

127

CHAPTER

GROUPS

ACTING

I

ON G R A P H S

GRAPHS

i.

By a g r a p h union

V vE

two m a p s of

X,

we m e a n a set

of t w o

sets

t,T:E ÷ V.

re, Te

are

respectively.

e

although

we a l s o

case

is c a l l e d

e

The e l e m e n t s

of

way,

and E = E(X), V

the e d g e s

are

called

of X.

given with the

For

initial

and terminal

An e d g e w i l l

usually

be d e p i c t e d

vertices

e ~ E,

the

vertices

of

,,~e

that

te = Te,

in w h i c h

V(S)

= S n V(X),

a 19o~.

S

X. of

X

we w r i t e

If for e a c h is a s u b g r a p h

For e a c h e d g e be t h o u g h t

E

a l l o w the p o s s i b i l i t y

= S n E(X). S

of

is g i v e n as the d i s j o i n t

c a l l e d the

,e

For a n y s u b s e t

t h e n we s a y

V = V(X)

that ¢

Let us fix a g r a p h ,

E(S)

X ~

and the e l e m e n t s

vertices e,

X

e

of

X

of as t r a v e l l i n g

respectively.

We

set

e c E(S)

we h a v e

and

re, Te E V(S),

of X.

we d e f i n e

formal

along

the r i g h t

e

te I = T ~ I = le,

symbols

e I , ~*,

to

w a y and the w r o n g

Te ~ = tC: = Te.

k

By a pat h

(1)

usually

P

P

:

in

is m e a n t

a finite

sequence,

V O , e ~ l , v l , . . . , eenn, Vn

abbreviated

le~.i = v i _ l ,

X

T e i~"

e1i,e22,...,e = vi

n,

where

for i = 1 , . . . , n .

n ~ 0, We s h a l l

e i = +1, call

and

v 0 the

GROUPS

initial

vertex

P is a p a t h

of

from

Two e l e m e n t s in

X

on

easily

Let

t_~o X

P be a p a t h

is,

X

say t h a t

in

X as in

If

we h a v e

a simple

e~l, "'''

called

the r e d u c e d

can be

s h o w n by i n d u c t i o n

a forest, there

reductions

vertices;

at l e a s t

and a c o n n e c t e d

By Z o r n ' s V(X')

of

at a v e r t e x

this

= V(X)

an e q u i v a l e n c e

class

of t h i s

relation

X.

if P

is c a l l e d

a component) ,

and

it is

We say t h a t

X

is c o n n e c t e d

We

P

is r e d u c e d

say t h a t

ei+ 1 : e i

is not r e d u c e d

of

reductions

by a s u i t a b l e

is, by the

is a p a t h

P

then then

ei+ 1 ~

for some

el+ 1 = -si; gives

-e i,

in t h i s

case

the p a t h

eei-1 =ei+2 .,een i-1 '~i+2 ''" n "

path,

of l e n g t h

if t h e r e

defines

(1).

reduction

simple

A circuit

and say

This

ei+ 1 : e i and

By s u c c e s s i v e

be f o l l o w e d

P,

component.

el+ 1 = e i.

two simple

of

of

n.

(or s i m p l y ,

i = 1,...,n-1,

i = 1,...,n-1,

vertex

s a i d to be c o n n e c t e d

An e q u i v a l e n c e

o n l y one

terminal

of l e n s t h

of them.

of

ON G R A P H S

the

n

vn

are

both

component

if for e a c h

itself

v

s e e n to be a s u b g r a p h

if it h a s

we

of

X.

a connected

and

v0

containing

relation

that

P,

ACTING

above, path will

we can t r a n s f o r m

f o r m of on the P

P. length

either

It is of

give

reduction

v

X

of I.

to a r e d u c e d

in fact u n i q u e ,

P,

equal

simple

P

noting paths,

to g i v e

is a r e d u c e d

that

path

is c a l l e d

a unique

reduced

be c a l l e d

a tree.

Lemma there

is a s u b g r a p h

X'

and m a x i m a l

w i t h the p r o p e r t y

of that

between X

v

to

is c a l l e d

In a t r e e

path between

a seodesic

can

paths.

from

A g r a p h w i t h no c i r c u i t s

forest

any

or each

equal

as

any p a i r of the v e r t i c e s .

having X'

is a forest.

GRAPHS

By m a x i m a l i t y , by an e d g e connected X',

of in

in w h i c h

maximal

no t w o c o n n e c t e d X,

section

components

vertices

In p a r t i c u l a r ,

if

case

X'

called

is a tree,

of the

assembled

by g i v i n g

For any r i n g

connected

all t h e s e

R

and

set

X

graph

in

we w r i t e

@ Rs, seS

= s.r

r

almost

• R,

s

1.1

R[S]

will

all

PROPOSITION.

Write

E = E(X),

(2)

(i)

by

so is

t r e e or a

for the set

r • R, ~ r .s SES s

this

S.

R-biThus

s • S. =

The

E s s•S "rs'

where

zero.

Let

R

be a n o n z e r o

V = V(X).

0 + R[E]

determined

for all

be e x p r e s s e d

then

of trees.

R[S]

R[S]

of

a l r e a d y be

let us c o n c l u d e

by the R - c e n t r a l i z i n g

elements

must

X.

freely generated w i t h r.s

X

characterization

S

can be j o i n e d

a spannin$

module =

X'

is c o n n e c t e d

definitions,

an a l g e b r a i c

of

connected

X'.

subtree

Having

so two

§I

There

~ ~ R[V]

rin~,

and

is a s e q u e n c e

e ~ R +

of

X

a graph. R-bimodules

0

(e)~

: le - Te,

sequence

is e x a c t

at

R[V]

if and o n l y

if

x

is

The

sequence

is e x a c t

at

R[E]

if and o n l y

if

X

is a

The

sequence

is e x a c t

if and o n l y

The

(v)e

= 1

(e e E,

v e V).

connected. (ii) forest. (iii)

this e v e n t , R-bimodule v e V, ~eodesic

for a n y v e r t e x right

X(v,v0) from

inverse

t__oo v0

of

X,

X(-,v0):R[V]

= ele I + ... v

v0

+ene n

in the t r e e

if

X

the m a p ÷ R[E]

where X.

is a tree. 3

has

determined

e~l,...,eenn

in

an by,

for

is the

GROUPS

Proof.

(i)

The

cokernel

ACTING

of

~

generators relations Thus X.

Coker Since

(ii),

8

is

R

is n o n z e r o ,

(iii)

If

R[C],

X

with

is a n o n z e r o

element

Conversely,

if

and

tree. eB

X

That

is,

this

verifies

2.

GRAPH

Let

V(m):V(F)

is the

presented R,

v E V;

f o r all

e e E.

set of c o m p o n e n t s

on

of

e

X

the

elel

component

case w h e r e

X,

X(-,Vo)

- X(Te,v0) inverse

some

is not e x a c t

then each

of

has

and t h e n

so (2)

to c o n s i d e r

is a r i g h t

then

of

:

+ en e n

+ "'" at

R[E].

of

X

X

is a

itself

is a

sends

X(le,Te) 3

circuit

:

as d e s i r e d ,

e. and

claims.

ANO C O V E R I N G S

be graphs. of s r a p h s

e

We use the t e r m s

~:F + X

E(a):E(F) of

F,

÷ E(X)

~(le)

isomorphism

is the d i s j o i n t which have

= 1(ae),

e(Te)

and a u t o m o r p h i s m

u n i o n of t w o m a p s

the p r o p e r t y = T(ee).

that

Thus

of g r a p h s

in the

{e E E(r)[

Te = v}.

way.

For any v e r t e x

(3)

C

centralize

le = Te

edges,

Ker ~

X(le,v0)

all the

÷ V(X),

for e a c h e d g e

natural

of

MORPHISMS

A morphism

that

R-bimodule

(i) f o l l o w s .

is a f o r e s t

to

X(-,vo)

F, X

where

for any e d g e

(: le - Te)

v

saying

no r e p e a t e d

it s u f f i c e s

Here,

is the

is n o t a f o r e s t

e el 1 '''''e~ n

tree,

ON G R A P H S

star(v)

v

of

=

r,

we d e f i n e

(e ~ E(F) ] le = v}

v

GRAPH

We say that F,

e

is locally

the induced map

define

locally

injective

inJective

ANO C O V E R I N G S

surjective

star(v)

and locally

isomorphism.

MORPHISMS

if for each vertex

÷ star(~v)

analogously.

surjective

For example,

§2

v

is surjective;

of

and we

If ~ is both locally

then

it is said to be a local

the m o r p h i s m s

O

->

b are both

2.1

local isomorphisms.

PROPOSITION.

morphism,

and

containing that

is,

Proof. F'

of

Notice

v

~v e:F'

Let

be a vertex

÷ X'

t h e n there

exactly

F'

v

P

some edge

one vertex

and

Any subtree

e in

If X

F' o f

F

a maximal ~:F'

If

X'

~

X'

graph of

X

containing

v,

+ X'

Since

of

e of

But as in

~v

X'

is a tree,

e

Thus

~:r ÷ x is a locally

is c o n n e c t e d

then

~

such that

e(F')

P

P

and has

is locally surjective,

F connected

r'

subgraph

is injective.

at

that does not lie in ~(F').

connected

is not an i s o m o r p h i s m

starting

e(F').

the m a x i m a l i t y

COROLLARY.

morphism,

in

in

we can find a preimage

2.2

such that

is then a tree.

is a path

contradicts

F.

Lemma there exists

does not lie e n t i r e l y traverses

of

surjective

is an isomorphism.

containing

that

be a locally

lifts back to a subtree

By Zorn's F

~:F + X

e

to

F',

and this

is an isomorphism.

surjective

is surjective.

graph S

D

GROUPS

Notice

that

subtree

T

subtree

F' of

T

of the

we can,

of

by

2.1,

and

such that

S,

of

S

of 2.2 we can c h o o s e

graph and

X

S

and

for e a c h

If • F'

choose This

Further,

are

S

of

S

by a m a x i m a l

subtree

of

S,

e

of

X

not

in

an e d g e

f

in

F

give

for

e (that

us a s u b s e t is,

the p r o p e r t y

all of w h o s e

we h a v e

be s a i d to be c o n n e c t e d .

a maximal

it b a c k to a

will

has

j o i n e d by a p a t h e

lift

edge

is a t r a n s v e r s a l

and f o r a n y e d g e will

ON G R A P H S

star(le),

is b i j e e t i v e ) .

two vertices

subset

connected

~f = e

r

a:S ~ X

in

F

situation

by considering

such that S

in the

ACTING

lee

that

terms

S;

the

are

such a

It is c l e a r w h a t

so we can s t a t e

any

we m e a n

foregoing

as

follows.

2.3

PROPOSITION.

morphism lifts

and

A local

S

universal

2.4

X

of

if

r

Let e

injective. a single

X

F,

w h i c h has

of

X.

subtree exists

The

By 2.2 any c o v e r i n g

o_ff

X

a connected

subtree.

connected covering

T

graph

D

graphs

is c a l l e d

is said to be is s u r j e e t i v e ;

can say m o r e .

be a t r e e

v

surjective

as m a x i m a l

between

Any coverin$

Suppose

and t h e r e F'

~:F ÷ X

is s u r j e c t i v e

vertex

be a l o c a l l y ~9Y maximal

of

is a tree.

PROPOSITION.

seen that

r'

or a c o v e r i n $

is a t r e e we

Proof.

e

isomorphism

a covering,

~:F + x

be c o n n e c t e d .

b a c k to a s u b t r e e

transversal

if

X

L~

and

of a t r e e

e:F + X

of

X.

be a c o v e r i n g .

so it r e m a i n s

that two vertices T h e n the

is an i s o m o r p h i s m .

of

to s h o w t h a t F

reduced

are m a p p e d

We h a v e e

is by

path between

~ them

to in

GROUP

F

is m a p p e d

is l o c a l l y zero.

injective.

This

is l o c a l l y Hence

3.

to a r e d u c e d

~

We

that

injective,

G

a set

X

as b e i n g

a

multiplication the

G-subset

same Ggx G/G x

by

g g,

stabilizer

under G/G x

Gg x are

Gx~ " =

{gGxl

:

is g i v e n

set of o r b i t s

G-action

v

to

v,

since

so the p a t h has on v e r t i c e s .

injective

length

Since

on e d g e s

also.

[]

(that

the

X,

Symx,

if

the

permutations

group

are

will

usually

be t h o u g h t

of as left

x ~

(x e X)°

For any

G

and the

orbit

of

X.

so the We the

of x Notice

write

hg

Choose

for

for

There

x • X,

in the

so

of G x in G,

correspondence of

with

Gx,

G-sets,

left m u l t i p l i c a t i o n . is a n a t u r a l

a transversal

is a t r a n s v e r s a l

g • G,

g~hg,

an i s o m o r p h i s m by

to be the

of t w o p o i n t s

set of l e f t c o s e t s

G-action G\X.

that

The

subgroup

is d e f i n e d

stabilizers

is a c t u a l l y

its n a t u r a l

~x

arguments.)

to be the

is in b i j e c t i v e

is d e n o t e d

to

on

of

of x is d e f i n e d

This

S

(As m a p p i n g s

and d e n o t e d

g • G},

G

acts

of t h e i r

For x • X,

is,

from

G

left

conjugate.

x ~ ~ = Gx.

or say t h a t

on the

gGx~1,

g G x ~-+ gx.

X ~ GXX,

X.

Gx = {~x I g • G}

orbit =

is i n j e c t i v e

homomorphism

written

G x = {g • G I ~x = x},

we h a v e

is a t r e e

G-set,

of

of an e l e m e n t

x • X,

from

it is t h e r e f o r e

a group

of all p e r m u t a t i o n s

image

X

X

be a g r o u p .

is g i v e n

viewed

~

in

ACTIONS

call

there

But

is an i s o m o r p h i s m .

GROUP

Let

shows

path

§3

ACTIONS

for

S

in

if The

surjection X

f o r the

X ÷ GXX).

Attach

to

GROUPS

each

element

of G \ X

G(~)

= G x.

Then

ACTING

ON

GRAPHS

of

G

by w r i t i n g ,

G-set,

X =

V ~G\X

want

a subgroup

as

In s o m e

situations

we

shall

the

right,

in w h i c h

case

X

set

of o r b i t s

the

use Let

of

is t h e n

G/G

X

be

a graph,

say t h a t

there

is g i v e n

graph

automorphisms

that

gae

G

acts

a group

= ~e,

g~e

EXAMPLE.

the (

(g,a)

E E(F)

multiplication edges.

Some

F(G,A)

Let

act

T(Ge)

= GTe,

There X + G\X,

as

on

some

a right

a notation

set

X

on

G-set.

which

all

The

generalizes

by

are

G-sets g E G.

G.

The

Cayley

: g,

T(g,a)

A

first

graph

= G×A,

F in a n a t u r a l

on t h e

if

a way

~w

1(g,a)

illustrated

of all

Pictorially,

E(F)

on

if

in s u c h

= G,

and

only

group

V(F)

acts

vertices

t o the

e ~ E,

of

E(X).

is a . G - g r a p h ,

are

gv ~¢

follows:

E =

G

V, E

=

G

X

from

a subset

if a n d

for

the

graph

maps

are

clearly

x ~ ~

S,

= ga way

components

on p. 13. generates

We G.

and

by

left

of the

remark

that

D

X.

incidence

is t h e n

)

given

examples

G

the

act

called

or t h a t

for

Here

connected

where

to

V = V(X),

Then

be

are

on the

G\X

X,

~

A

is

We w r i t e

on

~

).

on

write

= z~e

Y

Let

maps

and

X.

is d e f i n e d

incidence

X/G,

homomorphism

of

g(

F = F(G,A)

denoted"

X ~

G/G(~).

G be

each

above.

x

We

3.1

will

for

given

V(G\X) by

= G\V,

l(Ge)

E(G\X)

= G\E,

= G1e,

well-defined.

a natural : Gx.

with

For

surjective any

vertex

morphism v

of

of g r a p h s , X,

the

induced

map

GROUP

star(v) then

+ star(V)

for some

is the Gv,

image

acts

common some

on

initial

shows

either

gee

vertex

§3

for if

gle

star(v).

star(v).

g E G,

This

is s u r j e c t i v e ,

g ~ G of

ACTIONS

or Notice

Further,

if

and if

~i

and in fact

g E Gv

since

the n a t u r a l

map

gTe

equals

that

the

el,e2

v,

that

~ E star(V),

= ~2,

v,

so

stabilizer

of v,

are two e d g e s then

gel

gv = glel

Gv\Star(v)

e c E,

with

= e2

= Ie2

÷ star(V)

for

= v.

is

bijective. If back it

G\X

to a c o n n e c t e d S

that

is c o n n e c t e d ,

say. lies

element qe = 1. called

qe

of

same G

A family

element

~

(4)

of

of

G\X

G(e)

G-orbit

as

family

for

apply

X

for the

Te,

and we m a y

E S; )

),

G\X call

vertex choose

if

Te E S

chosen

in this

of

S

an

we c h o o s e way

is

S.

it a g r o u p

( e e E(S)

lift

G-action,

is a u n i q u e

G\X can be e x p r e s s e d with

2.3 to

there

~xe

( qe I e e E(S)

and we can a s s o c i a t e edge

in

e e E(S)

such that

a connecting

Each

transversal

For e a c h

in the

t h e n we may

uniquely

G(s)

we h a v e

'''~

as

= G s.

~,

Then

a commuting

s E S, for e a c h

diagram

[qe

~'~G(~)

~ G

,

namely /

Gle

~ G

GqeTe

~. G

g~

~ g

(5)

This

situation

will

n o w be a b s t r a c t e d

g ~"~gqe~.~ gqe and

studied.

10

GROUPS ACTING

4.

GRAPHS OF GROUPS

4.1

DEFINITION.

A graph

the f o l l o w i n g

data:

group

associated

~(y);

homomorphisms ~v),

~(e)

groups

(v

If

Y

we see that

e ~ E(Y)

of

~,

we call

e

)

with each of

Y

of a

y c Y

two group

÷ ~(~e).

The groups

are called the v e r t e x

respectively.

a graph

~

of groups ~:Y

÷

a connected

as a small

a graph of groups

the c a t e g o r y

We shall depict

an

to

Groups

(~,Y).

Groups,

We shall often this

its only d i s a d v a n t a g e

is the usual b e i n g that

of o t h e r a l g e b r a i c

(4) gives us a graph of groups

information

n o w see h o w to m a n u f a c t u r e

concerning

graph of groups.

4.2

DEFINITION.

Let

and

T

a maximal to

(~,Y)

subtree T,

with the f o l l o w i n g

G

a situation

from any c o n n e c t e d

~tSth r e s p e c t

way,

such as rings.

It is clear that extra

from a graph to

Indeed,

for graphs

D

in a certain n a t u r a l

is a f u n c t o r

for a graph of groups,

structures

category

graph of groups.

and group h o m o m o r p h i s m s .

it does not admit an analogue

certain

associated

Te:~(e)

consists

-~ Groups

as

By v i e w i n g

notation

~:Y

with each edge

~ V(Y),

is c o n n e c t e d

abbreviate

Y;

~ ~(te),

and edge groups

edge of ~

of groups

a graph

te:~(e)

ON GRAPHS

of

exactly

We shall

graph of groups, grou~

is the group that

associated

with

like that of (4)

The f u n d a m e n t a l

~ = ~(~, T), properties:

and the qe"

be a c o n n e c t e d Y.

together

w i t h each

of

is u n i v e r s a l v e V(Y)

11

GRAPHS there

is a g r o u p h o m o m o r p h i s m

edge

e

inner

automorphism

of

Y

(6)

there

~(e)

commute,

and

Thus

~

OF G R O U P S ~v)

associated

÷ z;

is an e l e m e n t

qe:~ ÷ ~,

§4

of

qe

makes

p ~ pqe

/

~

with

such that

each the

the d i a g r a m

[qe

such that

qe

is the g r o u p

= 1

presented

generators

e e E(T).

on:

the e l e m e n t s

generators

qe

the r e l a t i o n s relations

if

of

~(v)

( e c E(Y) of

~(v)

saying

( v e V(Y)

);

( v e V(Y)

(gte)qe

);

:

gT e

); for all

g e ~(e)

( e e E(Y) relations In the with

previous

connected

groups.

We

shall

4.3

need

the

~

as a left

~(v)

+ ~;

and

gives

on a g r a p h

e • E(T).

rise

with

acting

D on a g r a p h

to a c o n n e c t e d graph

connected

graph

of g r o u p s quotient

of

gives graph.

following.

setting

and r i g h t

for e a c h

composite

if

we saw h o w a g r o u p

graph

In the

view

qe : 1

n o w see h o w a c o n n e c t e d

acting

CONVENTION.

via the

section

quotient

rise to a g r o u p We shall

saying

);

~(e)

of 4.2,

~(v)-set

e ~ E(Y) + ~(te)

for e a c h

via the g i v e n

as a left

~ ~.

0

v • V(Y)

we

homomorphism

and r i g h t

~(e)-set

12

GROUPS

4.4 T

DEFINITION. a maximal

respect

to

subtree T,

V(F)

Let

=

of

:

since for any

graph of

~

with

is the graph with

E(F)

This

=

V ~/~(e), eeE(Y)

T(p~(e))

is easily ~g~(e)

=

pqe~(Te)

seen to be w e l l - d e f i n e d ,

= ~g%e~le)

section we shall see that

will be called the

standard

by 4.3.

F

0

is in fact a tree,

tree of

~

with respect

T. Notice

quotient

that graph

~

acts on

z\F

=

Y

F

by left m u l t i p l i c a t i o n ,

is connected.

Combined

and the

with remarks

§3 this gives us the following.

4.5 X

graph of groups and

are given by

g ~ ~(e),

In the next

of

The standard

p~(le),

( p ¢ ~, e e E(Y)).

to

Y.

maps

ON GRAPHS

be a c o n n e c t e d

V ~/~(v), veV(Y)

%(p~(e))

F

(~,Y)

F = F(~,T),

where the i n c i d e n c e

and so

ACTING

PROPOSITION so that

GXX

is q o n n e c t e d

transversal

S

connecting

family

in

graph of groups Conversely,

~ = ~(~,T)

X

for the G-action,

G:G\X if

G

acts on a graph

then for a n y choice

(qe I e e E(Y))

then for any choice group

If a group

(Serre [77]).

for

of c o n n e c t e d

and any choice S,

there

of

is a c o n n e c t e d

÷ Groups.

~:Y

÷

Groups

of m a x i m a l acting

is a c o n n e c t e d

subtree

on a graph

T

of

Y

F = r(~,T)

s r a p h of groups there

is a

so that

13

Group

D~

acting

on

1

:

of

groups

(ab) ~ = 1 >

I

a

ba

aba

- bab

C Cayley

Graph

graph

= < a,b I a 2 = b 2 =

§4

OF GROUPS

GRAPHS

=ab~ab

-

=

graph:

i©I D

....

ba

:

= < a,b I a 2

1

b

F Cayley

=

= b2

a


ab

=

1 >

aba

{l~b}

1~ { 1 , a }

>

graph:

D

1001 1 T "~

+

14

GROUPS We shall be examining the moment,

Y

~(~,T)

ON GRAPHS in detail

let us b r i e f l y m e n t i o n

from one-edge If

ACTING

in Chapter

the two basic

II;

for

cases arising

graphs.

has one edge

and two vertices

then a graph

of groups

can be depicted A

where

A, B, C

C + A,

¢

B

are groups

C ÷ B.

and there

Here the m a x i m a l

the fundamental

group

subtree

is presented

identifying

the two images

relations

of

A

and

B.

eoproduct

of

A

and

B

C + A,

C + B

of

shall not be using If

Y

of

group

A and this

has one edge

B

C,

and

with with the

AHB called the C If the h o m o m o r p h i s m s

C.

maps then

terminology

and one vertex

AvB

together

amalgamating

latter

graph,

is denoted

amalgamating

are given as inclusion

the free product

is the whole

on generators

relations

This

are given h o m o m o r p h i s m s

A[B is called C denoted A~B;

C,

we

and notation.

then the graph

of groups

can be d e p i c t e d

where

A,

Here the is

C

are

groups

given

maximal subtree

presented

relations be denoted

is

the

on g e n e r a t o r s

saying

~ea.q

HNN<~,B:C

since

it first

given

as an inclusion

with

vertex,

Av{q} =

cB

with

and the the

called

the

relations

and

6

÷ A.

of

group A

and

This group will

HNN construction ,

in H i g m a n - N e u m a n n - N e u m a n n

map

~,~:C

fundamental

for all e e C.

÷ A; q>,

appeared

two h o m o m o r p h i s m s

is injeetive

[49].

then

If ~ is

15

§5

A TREE

HNN<e,8:C

÷ A;

A~

and

5.

A TREE


Throughout (~,Y)

is called

q>

{:Cq

this

= C~,8>

Write

~

Our objective

more compact

although

this

of

~

graph

of groups

Y.

of

r = F(g,T).

now is to show that

notation

groups

T

denoted

Ac.

let us fix a connected

subtree

= ~(g,T),

variously

HNN extension,

and

section,

and a maximal

an

we shall

is d a n g e r o u s nor is

~y

write

since the

F

is a tree.

~y

in place

~(y)

the stabilizer

To make of

the

~(y),

y e Y,

are not given as sub-

of

y

in any genuine

sense. Let

R

suffices

be any n o n z e r o by i.I to verify

ring.

To show That

exactness

(7)

0 ÷ R[E(F)] .,% , R[V(F)]

determined

by

There ring, left

(e)~

= le - Te,

is a natural

R[~]-modules

(nonzero)

y ~ Y

R[V(F)]

we let

= i.

R-bimodule

R[~]

into a

is then a sequence

act t r i v i a l l y

homomorphism

of

denote

of

on The final

relations ~v'

~v ÷ ~'

where

the element

R[~]-module

generators

(it) left action

~

is the left (i)

the

R-bimodules

R + 0

(7)

and

of

it

t e r m of (7).

For each Then

E

(v)e

if we let

is a Tree,

of the sequence

way to make the

called the group ring,

F

~,

presented

of

F.

on

v e V(Y);

saying ~v

l~y

v

remains

fixed under

acts by pullback

along

the

16

GROUPS ACTING ON GRAPHS

The

left

~x]-module

R

(iii)

To see this, the

Q

are

the c o m m o n

in the

element

the

of

edges

of

e

for each (iii),

addition,

A

for all

Y.

edge and

is f i x e d

e

of

(ii),

T,

(iii)

by all

the m a p that

E 7.

~v

qe = 1 then

so all

say that

and all qe

that we

so is

that

is,

a

this

ad.b

ad m

that

without

~ ÷ R[E(F)]

brings

Let

Z[G]-bimodule. =

(7) is e x a c t

R[E(F)]

:G + M

derivation,

called

derivation

corresponds

a geodesic;

prior knowledge

is the

fact that

we an

the

of a g e o d e s i c .

for any v e r t e x

pIF(p2~,v)

+ F(p1~,v)

to h a v e

v

the

for all

important

concepts

with.

G

be a g r o u p

[g,m],

the i n n e r

1

v,w e V(F)

w i l l have

=

and

1 .

G-bimodule, is a map

For e x a m p l e ,

[g,m]

derivation

G

a

d:G ÷ M

a,b E G. where

M

=

by

of the

that

such t h a t if

gm - mg,

induced

to a g r o u p h o m o m o r p h i s m

(IdG 0

at R[V(F)].

is a tree,

p ~ F(p~,v)

for all

g ~

F

for e a c h

represents

A derivation

+ a.bd

is e x a c t

and

us to one of the m o s t

shall be w o r k i n g

DEFINITION.

(7)

(iii)

connected.

F(plp2~,v)

5.1

so the r e l a t i o n s

and t h e r e f o r e

us to do this

This

(e)~

to try and d e f i n e

of

will permit

=

3

proof

is to a c h i e v e

property

then

le = qe.Te

t5 - Te

by 1.1

F(v,w)

problem

(ab)d

saying

Q

F is t h e r e f o r e

are m o t i v a t e d

Pl,P~

=

cokernel

To c o m p l e t e

F

relations

by

of the

~e - qex~e

By 1.1,

of

with,in

7.

But

What

presented

that

identified image

f i x e d by

hold

recall

is then

m. form

m ~ M is a

A

17

§5

A TREE

If

M

is given

bimodule

with

satisfies sends

trivial

(ab)d

g

to

= (a)d

Let

derivations. There

for each

exists

be a

the

statements

~-bimodule,

following

a (unique)

e • E(T),

+ [ p l e , m e q ~]

In p a r t i c u l a r , derivation

a

G

a derivation m e M,

hold

ad m

for right

if and only

m e = 0;

the

V(Y)

that

the

v ¢ V(Y)

and

for each

e • E(Y),

=

Y ÷ M,

0

for all

y ~ my,

for each

ad m ~ M v~ ~d

dia$ram

p• w e .

there

e E E(Y),

is a (unique)

(qe)d

= me .

commutes,

if, me:

0;

(9)

for each

e E E(Y),

~e

centralizes

It is clear

qe

that

for each

e • E(T),

each

of

such

and

for each

There

a family

d:~ + M

(8)

(a')

)

)

are e q u i v a l e n t .

- ((PTe)dTe)~$

such

v E V(Y)

(m e • M I e • E(Y)

derivation

for any map

d:~ + M

for each

Proof.

it into

for any

(dv:~ v + M I v e

and

Then

For each

and

Here

also,

Analogous

e e E(Y), (qe)d = me, dv ~v .... M ~ /d commutes.

diagram

(ple)dle

can make

G-action.

+ a.(b)d;

M

f a m i. l y. .of. e.l e.m e.n t s ,

(b)

we

D

THEOREM.

(a)

G-module

right

(g-1)m.

G-modules.

5.2

as a left

that

is a (unique)

is sent

to

and

(a) is e q u i v a l e n t homomorphism

(qe qe),

qemTeqe

- m e q e - m~e.

to:

(01 d1 ):~ ÷ (0~ M)

and for e a c h

vertex

v

such of

Y,

that

18

GROUPS ACTING

I[ V

....

)

ON GRAPHS

"IT

1 d

(o (0 v M ) ~v

.........

,

~ M) (0

commutes. Now by the

universal

For e a c h

(b')

edge

property

e

of

T,

of m

w,

this

= 0,

is e q u i v a l e n t

and for each

edge

to:

of

e

e

Y,

i ~le) (0 e~

%

~le

(~le M ) ........~ (~ M ~le 0 ~ ) ( ~ e me)

4

(~Te M ) ~Te

qe IT

1 ~Te) (0

Te

commutes. condition

Performing says

+

-~ pl e .m e qe"

p T e . q e-I

(b) ,

and the

=

We now r e t u r n

-

q-I e'me'PTe

it follows

Y,

e el I ,...,e elln

=

v0 of

Y.

this

latter

PTedTe" (b')

is e q u i v a l e n t

of

(7).

define ^

ele I +

is the

:

that

the e x a c t n e s s

of

that

0

A

F(v,w)

Fix a vertex

v,w

^

shows

P ~ ~e'

is proved.

to p r o v i n g

For any v e r t i c e s

where

"* Ple, qe.

theorem

~ (~ M) 0

computation

for each

Since

^

the

that

(Ptedle)qe

to

~ ......

... + e n e n ,

geodesic For each

from

v

vertex

to v

w of

in the tree Y

and

define ^

(10)

r(pw0,O0)

^ ^

:

pF(~0,v)

+ r(v,v0)

e R[E(F)].

T.

p E ~v'

19

§5

A TREE

In other words,

the map

r

• + R[E(F)],

p ~

F

^ ^ (pv0,vQ),

is

V

defined

as the

inner d e r i v a t i o n

For each edge

(11)

e

of

F(qeQ0,O0)

=

Y,

= T$.)

We want

for all elements (8) and e

of

and

of

r,

in this

= 0,

that

qeF(O0,~e)q~ which,

by (11)

and this and

(10)

(11).

and

definition

(11)

(10),

v e V(Y),

so there

determined

for each edge

which

is easily e

have

~

that

verified

of

Y,

re

that

to be the case, centralizes

stabilizes)

- F(~0,~e)

of the right

by

is defined

F(_Oo,Vo)

r e,

÷

-i action, qe

as desired.

for all

p E r,

equals

Thus

5.2

and satisfies

^

+ ad F(v,v0)

is a w e l l - d e f i n e d

^

r(pv,vo)

It just remains map

F(pv0,O0)

R-linear

vanishes

on

rv

by

map

R[E(F)],

by ^

(12)

to define

(8) requires

stabilized

F(_,Q0):R[V(F)]

because

Here

^

For any

R[E(£)].

to verify

- r(qeO0,Oo)q~

F(pQ0,v0)

e

and by 5.2 it suffices

and the t r i v i a l i t y

is indeed

applies,

(10),

for each edge

case means

F(v0,v).

- ~ + £(t%,00)

logical

to combine

F(qeV0,V0)

(9) requires

$,

is the

(9) are satisfied. T,

(which

p

by

define

qeF(Q0,fe)

(It can be seen that this qe~e

induced

in (7). (p$)~

=

=

pr(O,O.)

to check that For any edge pCe - pqe~e,

A

+ r(pO0,v0),

F(_,0o) p~

of

p • r,

is a right F

and a p p l y i n g

v • V(Y).

inverse

(p • r, e • E(Y) F(_,v0)

of the )

we

then gives

20 GROUPS

r(p~e,Oo)

ACTING

ON GRAPUS

- r(pqeT%,0o)

=

pF(¢e,vo)

+ F(pOQ,Vo)

- pqeF(~e,vo)

:

pF(~e,Oo)

- pF(qeOo,Oo)

- F(pqeVo,Oo),

- pqeF(~e,Oo),

since

by

(12)

r(_Oo,Oo)

is a

derivation :

pe,

Thus by

by

r(_,0o)

1.1,

5.3

(11).

F

standard

p ¢ ~,

(Bass-Serre,

~:Y

÷

graph

We can

view

of

3,

so

Hence

is a tree.

THEOREM

~-set

(7) also

F r o m the

two exact

[77]).

and m a x i m a l

F as right

R[~]-modules.

5.4

Serre

Group8

F(~,T)

y ~ F.

following

inverse

(7)

is exact,

and,

is a tree.

THEOREM

of groups

is a right

For any c o n n e c t e d

subtree

definition

of

Y,

yp

= ply

the

D

by d e f i n i n g

gives

T

graph

an exact of

F

sequence

we t h e n

for all of right

have

the

sequences.

(Chiswell

[73],[76]).

The

sequence

of left

R[~]

modules (13)

0 +

@ R[~/~ ] B E(Y) e

determined

by

v ¢ V(Y)

is exact.

)

Duallx,

the

(P~e)B

= P~%e

se.quence

@ R[~/~ v] ~ V(Y) - pqe~Te ,

of r i g h t

R ÷ 0

(pnv)S

R[x]-modules

=

I (p e ~,

e ~ E(Y),

21

THE STRUCTURE

(14)

0 +

determined

@ R[~e\~] E(Y)

by

(p c 7, e e E(Y),

remarked using

of using the

by Chiswell;

the above

are n o w several

Serre

[77],

and perhaps properties touching

proofs

Chiswell

of universal

0

(~vp)C

sequence

=

to prove

follows

Dicks

upon.

of 5.3 available [77],

[79].

is Chiswell

[79],

coverings

that

From our viewpoint since

5.3 was

[77],

[79]

get

in the literature,

The most geometric, which

is based

our treatment

the above

we shall be needing

so this way we v i r t u a l l y

proof

will not

on be

is the most

5.2 for other purposes,

5.3 for free.

THE STRUCTURE THEOREM

We can now give a fairly between

6.1

÷

0

proof

[73],

the simplest,

appropriate

6.

exact

R

new notation.

There cf.

is exact.

§6

>

4 ~le p _ ~ Teqep,

v • V(Y))

The p o s s i b i l i t y

e

~ @ R[~v\~] V(Y)

:

(~ep)~

THEOREM

groups

THEOREM

Then there

the maps

graph

G-action,

and a maximal

~gfined

by

on graphs

(Bass-Serre,

on a c o n n e c t e d for the

acting

complete

T

of

is a c o n n e c t e d

G(~)

= Gs,

[77]).

Choose

a connecting

subtree

and graphs

Serre

X.

of the r e l a t i o n s h i p

of groups.

Let

a connected family

G

be a group,, acting

transversal

(qe I e e E(Y))

S

in

for

S,

S. graph o f groups

s e S,

G e ÷ Gte , G~$Te

picture

where

for each

are given by

G:G\X ÷

Group8

e e E(S),

g ~ g,gqe

~e_sspectively.

X

22 GROUPS

F r o m the g r o u P

ACTING

z = w(G,T)

ON G R A P H S

there

is a s u r j e c t i v e

homomorphism

^

+ G, G(~)

p ~ p,

÷ G

uniquely

to~ether

determined

with

F r o m the tree

q~ ~ qe

F = F(G,T)

by the

inclusion

(v E V(S), there

maps

e • E(S)).

is a u n i v e r s a l

coverin$

^

F + X,

pG(~)

~ ~s

(p • ~, s e ~)

that

respects

the

~roup

actions. Moreover,

F ÷ X

is b i j e c t i v e

if and only

if

w + G is

bij ect ive.

Proof.

The c o n s t r u c t i o n

the fact

that

G\X

The e x i s t e n c e the u n i v e r s a l of

S,

the

of

T.) The m a p

actions,

connecting

To

e E E(S),

p E w,

pG(e) ,

the m a p

this

with

~ + G

qe

certainly

the

star(1G(~))

from

(4) and

the

= 1

T

image

is a s u b t r e e

for all e d g e s

the g r o u p

of G(~)

{pG([) I pG(~-~)

= G(V)}

in

morphism,

in

G

notice

that

F

ye

isomorphism. to

star(v)

X.

Since show t h a t

We m u s t the

v {pG([)I

is an i s o m o r p h i s m .

pq[G(~-~)

= G(Q)}

now

~-action

for any

star(1G(~)) =

e

p q ~ G (T-e)

it s u f f i c e s ÷

is c l e a r since

in

'star'

4.5 and

edge

qe"

is a l o c a l

from

respects

it is a g r a p h

q%

pe

that

will h a v e

since

is c l e a r

is.

•

^

p%e •

that

X

(Notice

~ ~s

see that

^

commutes

~.

family

pG(~)

pG (i-e) ......

check

of

and is w e l l - d e f i n e d

is sent to

if

of the h o m o m o r p h i s m

property

s.

Group8

G:G\X +

is c o n n e c t e d

F ÷ X

stabilizes for any

of

v e V(S) Now

23 THE STRUCTURE THEOREM

{pG([) I t--~ = V, p E G(~)}

=

=

GCV){

i-~ = V}

IG(~)I

is m a p p e d

bijectively

G(V){~G(~)t

v

Gv{~e

which

s e e n to be

universal

and

since

covering.

Notice

that

only

S if

F

is a t r e e

as a m a p of sets,

~ ÷ G

and all

the

if

proved

6.2

THE S T R U C T U R E

a tree

then

Notice structure

the

full

We c o n c l u d e also true,

and

r + x

is a

is i n j e c t i v e

by 2.4,

b y 2.2.

is s i m p l y

or s u r j e c t i v e

property.

been verified.

and h e n c e

so is

if and

Thus

is

~ + G D

is a t r e e t h e n the u n i v e r s a l

THEOREM

(Bass-Serre,

Serre

is an i s o m o r p h i s m .

the o n l y

theorem

not u s e d the

is a l o c a l

covering

z + G.

Thus

we

following.

~ ~ G

that

F ÷ X

V G/G s . sES

claims have

X

= v}

it is s u r j e c t i v e

corresponding

£ ÷ X is an i s o m o r p h i s m have

(~E(G\X))

{~Te

by 5.3,

F ÷ X

÷

this

h a s the

In p a r t i c u l a r ,

Thus

In p a r t i c u l a r ,

is n o n e m p t y ,

surjeetive

I e e E(S),

star(v).

V ~IGCs) soS

Since

T-7 : V}

to

le = v}

isomorphism,

v {pG(e) I T--e = Q, pq[ e G(~)}

v

Gv{e I e ~ E(S),

is e a s i l y

§6

is the force

this

information fact t h a t of

X

F

needed

is c o n n e c t e d ,

to p r o v e

f r o m 6.1 and

is

the 5.3.

converse

the

a n d we h a v e

5.3.

s e c t i o n by n o t i n g

follows

If

D

about F

[77]).

of 6.2

is

24

GROUPS

7.

AN EXAMPLE:

To

the

following This

Serre

group

on t h e

=

upper

÷ H

z ~

az cz

'

SInH

the

E ][

ad

- bc

= 1}

half

of the

complex

plane,

I m z > 0},

(M6bius)

S I = {z E ~I

carries

us d e r i v e

group

{z E ~I

linear

(a b ) : H c d Let

of the

let

[77].

acts

fractional

GRAPHS

theorem,

{(ca b) i a , b , c , d

=

H

as

structure

description

SL2(Z)

ON

SL2[Z]

illustrate

classical

ACTING

transformations, + b + d

[z I = 1 }.

" Any

(a b c d)

element

of

SL2(Z)

to

{z e H I

_ac

Iz

- bd

c~ = d~1

= Ic~l

}

if

e2 ~ d2

Re z

ac

}

if

c 2 : d 2 : 1.

(15)

HI

{z ~ View

the

initial T

vertex

denote

the

show that we

L : {eiSI

arc

T

see t h a t

i set

the L

It f o l l o w s

that

from

is

with

of a l l

is t h e

intersect

is c l e a r

and

~

only for T

(15)

:

~ e s [} terminal

translates

geometric way

an

that

the

as a n o r i e n t e d

SL2(Z)

of

L

under

translate

of

of

T

with

SL2(Z).

L

in c o m m o n

realization

point

edge

1 .~3 ~ + i--~.

of a tree.

an endpoint

geometric only

p =

vertex

realization

it t o h a v e is t h e

- ½

Let We

From

(15)

can with

L.

of a graph.

on t h e

shal

imaginary

It

25

AN EXAMPLE:

§7

SL 2(Z)

!

I I I

-2

........

"1

.........

0

TRANSLATES

OF

1

Sin

H

UNDER

Z

SL2(Z)

!

I I I. 1

A

~L

-2

-1

1

0

TRANSLATES

OF

L

UNDER

SL2(Z)

2

26

GROUPS

axis i

is

are

curve T

i, L

in

to

and T

could

circuit

since

T

matrix

A : (

),

over

generated A

by

Thus

Z

acts

for this

L

one p o i n t

contain

TL : p

are

orders

2, 4

SL2(~)

any e l e m e n t by t h e s e

B

of ~ SL2(Z)

of

fixes

SL2(Z)L

to a

It r e m a i n s

algorithm,

any

to u p p e r t r i a n g u l a r SL2(Z)

can be t r a n s f o r m e d

and

so

SL2(Z)

is

1 -1 B = CA = (1 0 )"

where

@,

in

by the r o w o p e r a t i o n s

1)

A,B

is no c l o s e d

no c i r c u i t s .

operations,

by

containing

But a n y c i r c u i t

By the E u c l i d e a n

: (0

L

there

once.

element

has

of

so both

LuAL

is c o n n e c t e d ,

and

which

LoBL

completes

is a tree.

transversal

Computation

T

and

that

and

T

SL2(Z)

does not

So

Hence

can be t r a n s f o r m e d

Hence

that

contains

exactly

by a s u i t a b l e

and h e n c e

i,

are c o n n e c t e d . the p r o o f

L

L.

matrix

A,C

fixes

A : (10 -1) 0 "

along

C : (0 1 )

identity

ON G R A P H S

only translates

is c o n n e c t e d .

It f o l l o w s

to the

The

where

includes

2 × 2

Now

AL

be t r a n s l a t e d

s h o w that

form.

0 ~ H.

passing

that

ACTING

on the tree action,

shows

that

generated and

= C4 H C 6. c2

since,

from e a c h

two p o i n t s

by

T,

and

orbit,

and

6 respectively.

A

and

we h a v e

of

T,

seen that

it

orbit.

stabilizers

-I,

is a c o n n e c t e d

by our d e f i n i t i o n

f r o m any one

the

L

B

of

L,

~L = i

respectively

So by the

structure

and which have theorem,

27 FIXED POINTS

8.

§8

FIXED POINTS

Let

G

be a group a c t i n g on a tree,

We shall c o n s i d e r fixes

a v e r t e x of

situations

X.

where

X. G,

or some element

of

G,

The first of these will occur quite

frequently.

8.1

THEOREM.

Proof.

If

Let

subtree is-a

of

v X

of

is finite then

be a vertex containing

G-subtree

subtree

G

X

of

X

X'

edge of

Gv

of

and also the t e r m i n a l

G

extremities,

leaves

tree on w h i c h

arrive

The next results the t e c h n i q u e s [76].

8.2

DEFINITION.

(16)

{e ~

d e f i n e d by saying

G

at a single

X'.

acts.

carries size as

~

e~ 1 ~ e~ n

to a

extremitx

v e r t e x of e x a c t l y

one

the initial e x t r e m i t i e s , If

X'

has more than or d e l e t i n g

Continuing

cf.

G

acts.

[79],

Serre [77],

1.6.4,

D

many of and

o r d e r on the set

~E {÷i,-i} for each g e o d e s i c

edges,

in this way,

from Dunwoody

be the p a r t i a l

I eEE(X),

X' X'.

or terminal)

vertex on w h i c h

are a b s t r a c t e d

Let

G

is finite and

t o g e t h e r w i t h the a p p r o p r i a t e

b e i n g standard,

Bass

of

X'

all the initial e x t r e m i t i e s ,

all the t e r m i n a l

we e v e n t u a l l y

permutes

extremities

one v e r t e x t h e n d e l e t i n g

of

X.

be the smallest

Then

an (initial

or terminal)

The action of

a smaller

Gv.

a vertex of

X'

and of the same

X'

if it is an (initial

fixes

and let

the orbit

containing

X'.

X

since each e l e m e n t

Let us call a vertex of

of

G

e~l

..,een. '"

n

28

GROUPS For any e l e m e n t shifts

8.3

e

if

of

Let X

G

g

Suppose

any edge

of

X

e el I ,...,e ~n

for some

integer

e

if

fixes

moved

by

has

g

X,

G.

no edge

vertex

= w

e.

we say that

Then

w

and c o n s i d e r

~e El I

such that

of

shifts

some g,

of

of

is m o v e d

by

fixes

X.

the

geodesic

g

Let

If

then

by

path

g

so

moves

g

does

every

length

not

vertex

of the

shift

of

X,

e n = e. choose

e IEl ,...,e

geodesic

eel, " .,enn,gell, . ¢ e

..,ge~ n

reduced,

e~ n = gei¢l

is the

for if

geodesic

the m i n i m a l i t y

from of

i

8.4

THEOREM.

(a)

Each

(b)

No e l e m e n t

of

(c)

Either

fixes

X

G

there

such

that

subgroups

Proof.

Hence

G G

fixes shifts

some

Gel ~ of

G

by

~n

Ge2 whose

;

v

is the

le~ i

is

g

if

g

to m i n i m i s e

from

v

to

gv.

from

v

to

g2v.

n> I

shifts

the

Then It is

e2¢2 ' ... 'een-I n-i

and

%e~ n = gTe~l,

which

eI .

contradicts

D

are e q u i v a l e n t . a vertex any e d g e

vertex

is an " i n f i n i t e

(a)~>(b)

(a)+(b) ~ ( c ) .

of

a vertex

then

to

The f o l l o w i n g

element

be

is a r e d u c e d

Conversely,

is a path

ye~l

n.

e

1

gen-¢n ,.. .,geiei,el i ' .... een n

and

some

X.

of

e n = e.

and

g

i

fixed

g

e = ~1.

be an e l e m e n t

g

whieh

ON GRAPHS

and edge

if and only

Proof.

least

of

e s > ge e

LEMMA.

vertex

g

ACTING

of

union

X.

of X

X. or for e a c h

E

path" '''

of

vertex

w of

E

w,ell,vl,e22,... is a chain

is all

of

of p r o p e r G.

8.3.

Consider

any g e o d e s i c

in

X,

v0,e~l ..... e ~ n , v n

29

FIXED POINTS

Notice

that

either

G

:

G

v0 then there e~l... hg

exist

shifts

geodesic

el,

has

contradicting

Gv0 g

... ~ Gvi

here,

the

subgroup

this m e a n s

t h a t the

directed

system

Suppose Then

G

~ ...

n o w that

of the g e o d e s i c

vertex from

v v

of

G

does

v0

to

our path

w

not

there

v0

that

so by

i

t h a t this

n e e d not be u n i q u e

to

passes to

(a),

is a v e r t e x

v.

Hence

such that

through

n

Vn,

v

such that

there

is an " i n f i n i t e

Gv0 g GVl g ... constant.

vn

m > n,

G

Gv

m u s t be the

c G

vm

.

source

Hence

u G n vn

GVn : Gen g Gvn+1' (c) ~ ( a )

Let e

=

u G V(X) v

a n d now

(c)

:

G.

the g e o d e s i c

= G

from

of

so vn

v

to

Vm,

For e a c h

so

n,

follows.

is clear.

e

is the

say

e

away

from

us w r i t e

for the g e o d e s i c

is

For any

is the v e r t e x vm

Gvm

G

v0 in the s o u r c e

such that

so

For some

G

contained

is a l a r g e s t

v.

: . u G v. V(X) any v e r t e x of X.

stabilize

is not e v e n t u a l l y

there

closest

so

we m e a n

so is p r o p e r l y

from

X

and t h e n

is a g e o d e s i c

It f o l l o w s

(Although

inclusion,

Gv

chain

for if not

Gen,

... ,hge~n

, by w h i c h

w : v 0 , e ~ l ,vl,e e2 2 ,.. .

an a s c e n d i n g

h ~ GVn-

(b).

~ GVn.

; vn

is c l e a r l y u n i q u e . ) In p a r t i c u l a r ~ vi set of v e r t e x s t a b i l i z e r s u b g r o u p s forms a

under

does not c o n t a i n

vi

G

G

for any v e r t e x

path"

Gel ,

-

= en

... ,he~gl ,hge~l

a source

G

eI

g E Gv0

,e~n,he~Sn

or

§8

be an edge first

points v,

edge to

of

in the

v

denoted

G[e,v]

X.

For any v e r t e x geodesic

and w r i t e v ÷ e.

= {g E G I e ÷ g v ~ ,

from

e + v;

le

v

of

to

v

otherwise,

For any v e r t e x

X,

v

if

t h e n we e of

a n d let us t h e n w r i t e

points X

let

30 GROUPS ACTING ON GRAPHS G[e] = G[e,le]. 8.5 g

LEMMA.

The following properties

Let

e

an element of (i)

G.

(G[e,v])g

(ii)

G[e,v]

(iii)

G[e]

(iv)

be a n edge of

=

f

then

Ag ~ A

A

A,B

For any subgroup H-subset

of

g ~ G H

G

then

G[e]

Proof. from

to

g e G,

G[e] ~ G[e]g, For

if

e,f

are in different

G,

if

A ~ C

A-B

B~

A

we say

v,

g e G

of

A~B.

denoted

A

so write

If

A~B

G[e]

and

A ~ B.

If

e

of

of X,

X,

H-subset if

Ge

C

@ G[e,~lle],

of

G.

is finite

and is an almost right

X. by considering

the geodesic

G[e,v] ~ G[e,le],

is an almost right Gv-SUbset.

G[e,le]

D

is an almost right

we see from 8.5 (iv) that G[e]

orbits.

is finite then we say

for some right

v

g0 f = e

is called almost-right-invariant.

For any edge

v

such that

and write

for every vertex

so by 8.5 (ii), for any

go

B,

G,

For any vertex le

ge I < el}.

is a l m o s t - r i s h t - i n v a r i a n t ,

Gv-SUbset,

and

G.

for any

then

of

if

PROPOSITION.

8.6

of in

of

or

is almos ~ equal to

for every

X,

X,

u Geg 0

is almost contained

B ~A

of

~ LG[e,lf]

For any subsets A

Gv-SUbset

{go G I g~ I < e I

fG[e,lf] =

a vertex of

G[e,61v].

For any edge

G[e,Tf]

v

Then the f o l l o w i n g hold.

is a right =

X,

are readily verified.

Also,

so by 8.5 (i),

is almost-right-invariant.

[]

for the cyclic subgroup generated by

g.

31

TREES

8.7

LEMMA.

then

G[e]

Proof.

If an element

so

g

has

g

-subset.

by

if and only

If

every

H

fixes

a vertex of

X

shifts

an edge

right

H-subset.

g[e]

then e,

8.4

G[e]

G[e]

of

X

we may assume contains

gn, n > 0,

is not an almost right

ordered

subgroups

o_~f G

fixes

is a m almost right

X

so by 8.7,

are finite

a vertex

Conversely,

if

G[e]

X

H-subset.

H

G[e]

does not

so some e l e m e n t

this

of

of

then by 8.6 every

(c) fails,

of

fix

H

is not an almost

D

ORDERS

section we a s s o c i a t e d w i t h

set

(16) p o s s e s s i n g

e e ~ e -e.

Notice

is altered then the r e s u l t i n g is v i r t u a l l y

c o v e r thereof.

that

Following

characterization

of those

X

if the o r i e n t a t i o n

Thus

tree,

a tree

a

a natural order-reversing

partially

the same.

the edge set of an u n o r i e n t e d

abstract

so

H

H-subset.

In the p r e c e d i n g

double

e

-subset.

a vertex of

T R E E S AND PARTIAL

involution

an edge

if n e c e s s a r y ,

gn, n < 0,

if

right

involution

shifts

If the e d g e s t a b i l i z e r

is an almost

partially

G

o r d e r and

b o u n d e d o r d e r then a subgroup

9.

§g

ORDERS

D

THEOREM.

Proof.

of

~I

infinite

and does not c o n t a i n

8.8

g

is not an a l m o s t right

On r e p l a c i n g

e I > ge I

AND PARTIAL

ordered

of

X

set with

(16) may be viewed as

or more

Dunwoody

precisely,

a

[79] we shall give

partially

ordered

sets w i t h

an

32

GROUPS

involution Let

which

(E,~,*)

involution exactly with

e E

for

f,f*. edge

We

set

write

e > f*

(In

interval

and

distinct.

no

are

Hence

e,f

shall

of

[f*,e] e ~ f

e

is

then

comparable e

to

g

contradicts

So

e > g*,

In

interval

comparable [g*,e]

e E

to

then to

with to

an u n o r i e n t e d

forest

E

= f

or

as

follows.

e

covers

strictly

= {f*,e}.)

To

f ~ g,

f*,

between

see

~

suppose

e,f*.

If a n y

e

exactly f*

one

cannot If

being

to

f

= [g*,f*] = {e,g*}.

or

the f*

u (f*,e]

two

e,f,g

of

be

e ~ g*

comparable

g,g*.

covered then to

If by

e

both

is e,g.

f* < e ~ g*

exactly

one

fact

that

every

element

of

of

E

f,f*.

is

says u [g*,f)

u If,e]

=

So

< f

as d e s i r e d .

notation

that

is

e , f , g E E.

so we m a y

For

f > g*.

g*. f

on

lies

g

is c o m p a r a b l e

which

~

e ~ g

e > f*,

comparable

set

e is c o m p a r a b l e

construct

and

f

Now

ordered

e E,

relation

element

equal

way.

partially

if e i t h e r

suppose

GRAPHS

E.

notation,

transitive, e,f,g

e ~ f

ON

in t h i s

any

an e q u i v a l e n c e

is,

of

that

of

(double)

arise

be a n o n e m p t y

such

one

Define e,f

can

ACTING

(f*,e] u [g*,f)

33

TREES AND PARTIAL So

e

covers

g*.

§9

ORDERS

It is n o w clear that

~ is an e q u i v a l e n c e

relation. Let is,

X'

{

edge

be the u n o r i e n t e d

{e,e*} {e,e*}

I e e E},

g r a p h with edge set

and with v e r t e x

has as v e r t i c e s

where we v i e w

vertex the class of

e e

E/~,

where the

of

e,e*

in

the classes

For any u n r e f i n a b l e is,

ei

covers

as the o r i e n t e d in

arises

for any

X',

forest.

condition

e,f e E

edge with terminal

e I > e 2 > ... > e n

and c o n v e r s e l y ,

in this way.

so is an u n o r i e n t e d sufficient

of

in

E

(that

{e n, e*n }

is a r e d u c e d p a t h in X'

orientations

i : 1,...,n-1),

~" {el,e [}

in

E/~.

E/~.

chain

ei+l,

that

set

We can t h i n k of e,e* as being the two p o s s i b l e {e,e*}

E/*,

for

the

X'

This

e v e r y reduced path

shows that

X'

has no circuits

It is c l e a r that the n e c e s s a r y to be an u n o r i e n t e d

(totally ordered)

interval

and

tree is that [e,f]

is

finite. This gives the c l a i m e d For our p u r p o s e s

characterization.

we need a genuine

can obtain this e i t h e r by t a k i n g a b a r y c e n t r i c

by choosing

(oriented)

tree and we

an o r i e n t a t i o n

subdivision

of

X'.

of

X'

or

It is the l a t t e r

that will meet our needs. Let where Te tree

X

be the graph with

for each

e E E,

is the class of it is c l e a r that

e

V(X)

=

E/* v E/~,

le

is the class of

in

E/~.

X is a tree.

Since

X'

e

E(X) in

=

E/*

E, and

is an u n o r i e n t e d

For any u n r e f i n a b l e

chain

34 GROUPS

e I > e 2 > ... > e n

in

ACTING

E,

71 71 72 ?2 is a r e d u c e d Let the inherited

Using

9.1

8.5

path

set

X,

say.

Then

the p a r t i a l

for any

in

E,

f-i < e I

if and only

if

f* < e

in

E.

(iii)

we now h a v e

(Dunwoody

of

an d the

E,

a n d the

to an a c t i o n

e

G

(E, ~, *)

ordered)

G

on

{g e G I ge < e

X or

acts then

in

E,

under

tree

on

E

ge* < e}.

a way D

~

s u c h that

to e x a c t l y

[e,f]

one

is finite.

X.

r e s p e c t i n $ the p a r t i a l

the a c t i o n

in such

be a n o n e m p t y

involution

interval

set of a c e r t a i n

involution, of

Let

is c o m p a r a b l e

a group

e,f

order

following.

order-reversing

(totally

is the edge

the

[79]).

set w i t h

in

If, m o r e o v e r ,

=

s

have

e s f*

e,f

G[e]

denoted

}

if

for e a c h

order

X.

if and only

ordered

E

en

e I < f-1

partially

Then

e~

{ee I e e E, E ~ {+1,-1}

from

THEOREM

f,f*,

in

ON GRAPHS

of

that

G

on

for each

E

extends e

in

E,

CHAPTER

II

FUNDAMENTAL

In m a n y given

group

acts, the

occurring

can be used

and t h e n the

group

This

naturally

situations,

to c o n s t r u c t

structure

as the f u n d a m e n t a l

can only

be u s e f u l

group

and the p u r p o s e

some

of the

salient

features.

here

are

of the

fundamental

substantially the

full

that

I.

force

the real

group

on the

Y

information constructed

that

assigns

edge

of

with

Y

Y:Y ÷

the

respect

< qe'

T,

group of

the is to i n d i c a t e

all the r e s u l t s

makes

the a c t i o n

the p r o o f s

is w h e r e

a tree.

original

a

of groups.

of u s i n g

tree

being

in the

one

applies

The m o r a l

action

is

of our g i v e n

GROUPS

graph

and

T

of

Y

~(Y,T),

I

qe

has

graph

the t r i v i a l

homomorphisms.

IE(Y)

be a m a x i m a l

for the t r i v i a l

element

e c E(Y)

so is free of rank

lies

- FREE

identity to

graph

graph

about

almost

this

the

a description

chapter

approach

about

tree.

Groups

to e a c h

of this

Indeed

standard

be a c o n n e c t e d

We w r i t e

is k n o w n

on the s t a n d a r d

transparent.

CASE

provides

Although

Bass-Serre

group

of the

THE T R I V I A L

Let

Y

more

the

on w h i c h

of a c e r t a i n

if s o m e t h i n g

group,

information

a tree

theorem

fundamental

classical,

GROUPS

The

of g r o u p s group,

of

and to e a c h

fundamental

group

>

In p a r t i c u l a r ,

if

Y.

o__n Y

presentation

= 1, e E E(T)

- E(T) I.

subtree

E(T)

is

of

86 FUNDAMENTAL GROUPS

II

finite then

IE(T) I = IV(Y)I

IE(Y) I - IV(Y)I bouquet

+ i.

of loops,

any c a r d i n a l

- I

Thus if

then

Y

~(Y,T)

can occur here,

and

~(Y,T)

is free of rank

has o n l y one vertex,

is free of rank

we deduce that

and only if it is the f u n d a m e n t a l

so is a

IE(Y) I.

a group

Since

is free

group of a trivial

if

connected

graph of groups. The trivial

graph of groups

on

Y

arises

when we a p p l y 1.4.5

to the a c t i o n of the trivial group on

Y.

is a u n i v e r s a l

covering

also

F(Y,T),

a group is said to act freely on a set if e v e r y

where

point has t r i v i a l We have thus a trivial tree;

1.1

stabilizer.

connected

holds

(Reidemeister

only if it act s f r e e l y G = ~(G\X,T)

Further,

graph of groups

THEOREM

is a~ain

Proof.

(Schreier

free.

Let

G

[27]).

A group

subtree

rank H

(G:H)(

be free on a set set

E.

G

X say. T

freely on some at the

is free if and In this event,

of

A subgroup rank G

of loops with edge

group of

G\X.

D

of this.

If m o r e o v e r =

= Y.

and we have a r r i v e d

on some tree,

Let us record one c o n s e q u e n c e

1.2

acts freely on

~(Y,T)kF(Y,T)

it acts

there

theorems.

[32]).

for any m a x i m a l

~(Y,T)

is the f u n d a m e n t a l

then

by 1.6.2,

form of the B a s s - S e r r e

THEOREM

+ Y;

shown that if a group

the converse

oldest

F(Y,T)

T h e n by 1.6.i

E,

and

H

of a free ~roup

(G:H)

rank G

then

- 1) + 1.

and let

Then the maximal

are finite

G

Y

be the bouquet

subtree

T

of

Y

is

37

THE TRIVIAL

just the u n i q u e v e r t e x

of

CASE

Y,

and

G = n(Y,T)

so acts f r e e l y on the tree

X = F(Y,T).

on

X

If m o r e o v e r

as

H-set,

so is free by 1.1. HXX

finite then rank

Y = GXX

H

Combining useful

1.8

is (G:H)

trivial

of

[E(H\X) I

=

(G:H)IE(G\X) I

:

(G:H)(rank

(Bass-Serre,

srouP,

and

H

G\X,

-

G

-

+

acts

is finite

so if rank

freely

then

G

is also

1

(G:H) I V ( G \ X ) I

- 1)

+ 1.

+

1

D

of C h a p t e r

and in fact

Serre

[77]).

i we get a very

Let

any subgroup

of

with each c o n j q s a t e

p~(v)p

(p e z(~,T),

~:Y +

z(~,T)

be

which has

of the image of each

v E V(Y)).

H = ~(HXF(~,T),T')

Groups

Then

for any m a x i m a l

H

!~

subtree

T'

H\F(~,T).

The h y p o t h e s i s

tree

~(~,T) .

1.4

COROLLARY

~(v)

ensures

that

H

acts

freely on the

0

(Bass-Serre,

is t o r s i o n

free.

Serre

[77]).

If each v e r t e x

then everz....torsion-free sub..~roup o f

grou[

~(~,T)

is

D

REMARKS

(Bass-Serre,

with a group of

(G:H)

H

way

and

1.1 with the c o n c e p t s

intersection

Proof.

1.5

Clearly

IV(H\X) I

=

.a. .graph . . . . . .of . . .@roups

free,

is finite

of

in a n a t u r a l

result.

THEOREM

vertex

copies

§I

- FREE G R O U P S

~(G,T)

~ G

G

acting

Serre

[77]).

on a c o n n e c t e d

satisfies

In the setting graph

the h y p o t h e s i s

X,

of 1.3

of 1.6.1

the kernel and further,

N

38 FUNDAMENTAL

II

N\F(G,T)

= X

1 + ~(X,T')

in a n a t u r a l + ~(G,T)

Let us n o t e we h a v e normal This

a surjection

says that

a surjection generated

2.

maximal

of

Y

will

group

on a t r e e

and the k e r n e l

and

of

Groups

groups.

then there

is the of

X.

is t h e

vertex X

of the v e r t i c e s

a vertex

way

precise

DEFINITION. will

of

subgroup X.

is of

G

0

that

we s h a l l

consist

following

a family

(tv:~V)

e

that

specialization;

theoretic

notion

but we s h a l l

not

the c o n n e c t i o n .

G,

(tel

call

functors,

For any g r o u p

a family

be a

s t u d y of the

f r o m the c a t e g o r y between

T

Y.

in o u r i n i t i a l

is a c o n c e p t

transformation

to m a k e

t : ~ ÷ G,

v0

be u s e f u l

in a c e r t a i n

w i t h the p r o p e r t y

(1)

of the

Groups be a c o n n e c t e d g r a p h of g r o u p s ,

subtree

of a n a t u r a l

G,

+

a n d the k e r n e l

images

acts

~:Y

T'

RESULTS

fundamental

2.1

G

sequence

subtree

of g r o u p s

+ ~(Y,T)

stabilizers

an e x a c t

for any maximal

by the

if a g r o u p

A tool that

digress

~(~,T)

G ÷ ~(G\X,T)

~:Y ÷

it a r i s e s

so we h a v e

for any g r a p h

generated

by the

BASIC

Let

+ G ÷ 1,

also that

subgroup

way,

GROUPS

of the

e E(Y)) + GI v for e a c h

a specialization

of e l e m e n t s E V(Y)) edge

data: of

G,

of h o m o m o r p h i s m s , of

e

~'(re)

,,* G

N('re)

It ~ G

~(e) /

from

e

Y

~

to

39

RESULTS

BASIC

commutes,

where

as usual.

the

: qe

2.2

there

for e v e r y

PROPOSITION

v

to

=

edge

tree

such

that

qe

~

I

1,

and

T.

edge

for e a c h

[77]).

Then

there

with

Y

t:~÷

G

be a

geodesic

from

define

is ..........the ..

is a u n i q u e

for each

e

of e

T,

edge

homomorphism

e

of

of

Y,

an__jd

The

result

vertex

v

=

of

T(le~v0) , G

therefore

T(vo , l e ) t e T ( T e , v o ) G

follows

from the

universal

~(~,T).

DEFINITION.

Let

~

specialization,

that

generators

e e E(Y))

(tel

is,

+ U(~) U(~)

denote is the

together

the u n i v e r s a l group

with

the

presented ~(v),

Y.

T(v0,v0)

Y,

; e , G T(Te~v~)~

9¢(Te)

for e a c h

T(v0,1e)teT(Te,v0)

~t

of

~(~,T)

Let

v,w of

commutes

edge

~(e)

commutes.

t:~+

e el I ,...,e~ n

I

~(le)-----+ G

2.3

Serre

' G

For each

property

automorphism,

~ G

~(~,T)

=

inner

Y.

T(v°'le)teT(Te'v°)

~(v)

Proof.

of

where

in the

+ G

e

For any v e r t i c e s

t:~...tgnen

w

~(~,T)

is the

is a s p e c i a l i z a t i o n

(Bass-Serre,

specialization. T(v,w)

arrow

D

For e x a m p l e , te

vertical

§2

on

v e V(Y),

40

II

FUNDAMENTAL gROUPS

and relations

consisting

with r e l a t i o n s

saying

For any path U(~)

that

where

for each

path

in

The

of the relations

that

for each edge

v0,e11,...,Vn_1,e~n,Vn

can be expressed

U(~)

in the

i,

gi

comes

from

v0

t__oo v n.

in

U(~)

set of paths

subgroup

of

U(~),

denoted

~(~,v0).

F(~,v 0)

defined

of the

from

from

is called

of

Y,

(1) commutes.

in

Y,

any element

of

form

g^tSlg, tS2...tengn, 0 eI i e2 en {(vi), will be called a

v0

acts

together

e

to itself,

the fundamental

This group

~(v)

clearly

group

in a natural

of

~

a at v0,

way on the graph

as follows:

V( F(~,v 0)

)

:

{P~(v) I veV(Y),

P a path

in U(~)

from v0 to v]

E(F(~,v0)

)

=

{P~(e) I eeE(Y),

P a path

in U(~)

from v0 to re}

P~(le) with

P~(e) ~

edges

where

the convention

For example, group

of

Y

fundamental

at

Ptef~(Te) ~ ,

similar

if

~ : Y

v0

reduces

to 1.4.3

is trivial,

of a connected

By 2.2 we have

a homomorphism

a homomorphism 'fixing' mapped

each

~(~,v0).

U(~) ~(v).

+ ~(~,T)

straightforward ~(~,v 0) + ~(~,T)

topological ~(~,T)

sending

In particular,

and therefore

notion

D

fundamental of the

space.

+ U(~)

and its image

But in the other d i r e c t i o n

to qe so the composite

identity,

then the

to the usual

group

be seen to lie in

is understood.

so is

~(~,T) ~(~,T)

each

te

each

T(v0,1e)teT(Te,v0)

÷ U({)

to

qe

÷ ~({,T)

+ ~(~,v0)

is the i d e n t i t y

is

is the

÷ ~(~,T).

so we have

is

and

to check that the composite ÷ ~(~,v0)

there

can

an

It is

4'1

§2

BASIC RESULTS isomorphism

~(~,T)

= ~(~,v0).

It is now a simple matter to

obtain the following. 2.4

THEOREM

(Bass-Serre,

of compatible

Serre [77]).

isomorRhisms

In particular,

~(~,T)

the isomorphism

There is a natural p a i l

= ~(~,v0),

subtree

T

of

Y.

= F(~,v0).

class of the fundamental

~cting on the standard graph is independent maximal

F(g,T)

group

of the choice of

D

Let us now consider the p r o b l e m of when all the h o m o m o r p h i s m s ~(y) + ~(~,T),

yeY,

are injective.

this to hold is that e

of

Y,

'faithful'

LEMMA. say

Let

G

condition

÷ ~(le), g(Te)

standard

seems preferable

to the correct

If

is also

approach that is

fact.

be a group actin$ freely on a set

e,B:G ÷ Sym X.

are

category theoretic

To see that this condition

elementary

for

for each edge

we take the less usual, but simpler,

based on the following

ways,

violates

'monopreserving'.)

sufficient

that is,

le,Te:g(e)

(This terminology

but here

'mono' or

2.5

be ~aithful,

the h o m o m o r p h i s m s

injective. usage,

g

A necessary

IGe\XI

= IGS\XI

X

in two

then there exists

Sym X !

_a

t E Sym X

such that

G

It

+ SymX

"~B Proof.

We write

respectively.

X,BX

to denote the

The fact that

~ X is the disjoint union of

G

left

G-set determined by

acts freely on

IG\ X I

G-sets

which is isomorphic to the (left) G-set are isomorphic

commutes.

G.

X

means that

(orbits) Thus

~,B

each of X

and

G-sets and we may choose an i s o m o r p h i s m

8X

42 II

FUNDAMENTAL GROUPS

t:BX + X .

Then

t(gB.x)

: g~.tx

for all

g ~ G,

2.6

~

resulting ~(v)

Proof.

on

actions

X. of

the number there

÷ Sym X

I~(v) l

Y,

on

a

X

X,

property.

e

is the same

gB

if

X

IX1

!~

is infinite.

such that

in the

the composite

that

Y

v

of

Y.

for each vertex

v

a free action

we then have on

X

for both actions.

of

two free ensures Thus

that

by 2.5

such that

> Sym X It e

gives

provided

, Sym X

......

the desired by 2.2

specialization,

clearly

since

the

has the requisite

[]

to satisfy

following.

divides

for each vertex

of

It is clear that we can always large

IXI

and the hypothesis

t e ~ Sym X

=

be a set such

IXI, so we may choose

x~'~Te)

homomorphism

X

~ ÷ Sym X

ensures

~(e)

This

than

÷ Sym X,

/~le)

commutes.

tJ.g~.t

I~(v) I either

is injective on

such that

is,

and let

smaller

~(~,T)

For each edge

of orbits

exists

2!

divides

~(e)

That

a specializ@tion

The hypothesis

Y,

~(v)

v

homomorphism

+ ~(~,T)

xeX.

Sym X

D

or is strictly

exists

of

be faithful

for each vertex

Then there

gEG,

as desired.

Let

X is finite,

of

is an element

for all

THEOREM.

that

t

the conditions

choose

of 2.6,

a set

X

sufficiently

so we have proved

the

43 THE FAITHFUL CASE

2.7 Y

COROLLARY. the

2.8

on trees,

connecting connected

T

graphs

of groups

be a maximal

v

given with

of

of

Y

of

we write

te,Te:G e + G e,GTe

Ge I ÷ G +1.

= G,

G

If

e

Gv

with

Ge 1, G +1 e

and a

subtree,

graph

by

faithful (~,Y,T).

of groups

T

Ge

with

map

G

G

Notice

determines

for

Y ÷ G

v

and

~(y).

is injective,

in

for the image

qe

in

write

its image

respectively.

induced lies

y E Y

the canonical

edge

of

a maximal

faithful

and for each

and we m a y identify

automorphism

transversal

groups

Y.

by 2.7,

Y

us that

and on the other hand,

be a connected

G = ~(~,T)

e

tells

on the one hand,

a connected

of

D

last result

between,

v

CASE

subtree

For each vertex

is injective.

I.§6 this

(G,X,S,(qe)) ,

~ : Y + Groups

Write

with

equivalence

family,

then for each vertex

÷ ~(~,T)

given with

THE FAITHFUL

Let

is faithful

~(v)

Together

is a natural

acting

3.

~

homomorphism

REMARK.

there

If

§3

G.

of

G

For each under

e

that the inner an i s o m o r p h i s m

then this map is the identity,

and

e

here we shall Let

identify

X = F(~,T).

Ge I = G +1. e

As in 1.§5,

there

is a canonical

embedding

^

T + X, t ~ t. preceding

3.1

Let us identify

conventions

PROPOSITION.

T

with

are consistent

For any edge

e

its image

with

of

stabilizer

Y,

Glen

- -

~nd for any vertices the

interseclion

edges

e

thai

v, w

o_~f Y,

of the edge groups

lie in the geodesic

in

X,

notation.

G q~

=

Te

Gv n Gw Ge from

so the

G -i e

equals

corresponding

to the

v

T.

t__oo w

in

'

Q

44 II

FUNDAMENTAL GROUPS

Proof.

In the action

element

of

it fixes

G

fixes

the

Gv

and

qe

g

in

T

we may choose path path

P

can reduce

X

m T

if and only if

D

generate

en "'" qengn '

G~

of

any element

G.

Since

g

of

G

gi e Gvi,

and edges

expression

i,

n,

so that g

in

Y

(2) to a path

By using

from the vertex

groups

v0,e~l,...,e~n,v n

is a

has been e x p r e s s e d

in

with underlying

e~i# 1 = e?Si i+I l

the expression

el,...,e n.

elements

We then say that

of !en~th

If for some

of

it is clear that an

form for the elements

and the identity

Y.

format

them.

v0,...,v n

this

in

V,w

X,

as a product

for some vertices geodesics

a normal

~I g0qelgl

=

on the tree

between

together

can be e x p r e s s e d

(2)

G

two vertices

the geodesic

We now c o n s i d e r

of

path

P.

and

gi ~ Get t h e n we ei format of length n-2,

el .qei-l~,qCi+2 en g0qelgl "" ei_l ~ e i + 2 " ' ' q e n g n where

g'

g i - 1 " q eeiig i q e-ei i "gi+1

=

On the other hand, the path format element course,

of

G

can be e x p r e s s e d

the u n d e r l y i n g

Lemma.

[63]

Gvi_ I = G%e [i = G vi+ 1 .

does not happen

(2) is said to be reduced.

We can n o w extend Britton

if this

•

for the

path

in reduced

does not have

to fundamental

groups

HNN construction

for any

i

then

It is clear that any path format.

(Of

to be reduced.) a result

and known

proved

by

as B r i t t o n ' s

45 THE FAITHFUL

3.2

THEOREM.

Gv0

can be e x p r e s s e d

underlying

be a vertex

u niquel~ v0

pat h format,

Let

reduced

v0

path is from

the trivial Proof.

Let

g

v 0.

of length

X

we have

1Gv0

a path

of

from

g qSl G 0 eI vI

Y.

Every path

element

format

This unique

of

where

expression

the is

0. G

path fore,at w i t h u n d e r l y i n g

Then in

of

in a reduced

to

be an element

§3

CASE

and suppose (2) is a v0 path from v 0 to itself.

v 0 = 1Gv0

to itself:

g qel~ nE2G O e1~1~e2 v 2

gGvn . o .

But

X

is a tree

inverse. that

of

On e x a m i n i n g

~i+l-ei+l = e?eil

the path

3.3

so there must be some edge

format

LEMMA. Gv,

in

being

generate

Proof.

G

Y

edge we find there

and

=

gi • Ggi'ei

reduced.

For each vertex

and suppose

(Hie n Gel) qe

this

followed is an

But this

v

o_ff Y,

If the

Hv

le t e

Hv

of

and

then

Hv = Gv

for each vertex

For each

e E E(Y)

write

Y, qe

v

H e = Hie n

be a subgroup

together of

,

Y.

and set

V G/Hy made a graph with the (well-defined) yeY relations l(gH e) = gH e , T(gH e) = gqeHTe . Since G,

X'

is connected

the image of a certain locally ensure by 2.4,

surjective that so

standard

(cf p.16, tree).

graph m o r p h i s m

it is locally Hy = Gy

or observe There

X' + X,

injective,

for all

such

contradicts

X' =

generate

i

[]

that for each edge

HTe n G e+1 "

by its

y e Y.

incidence the that

Hv,q e X'

is an obvious

and our hypotheses

so it is an isomorphism []

is

46

II

3.4

If

COROLLARY.

generated

and

G

generated.

Proof.

Take

finite each

by t h o s e

Gv = H v

image

let

be the

of

Y

generators

each

G

with

that

for each edge

e

of

beginning

section with

we

qe'

edge

subgroup G

of this

vertex

the

of each

for e a c h

is f i n i t e l y group

is

f r o m the g e n e r a t i n g

of

generated

group

and add in

group.

of

Gv

lie in Y,

vertex

shall

generated G v.

so by v

be

of

-

Then

3.3, Y.

looking

the two e x t r e m e s

For

D

at the

finite

index

finite.

3.5

THEOREM.

in some

If

conjugate

Proof. H

G,

Hv

edge

then

for

for each

For the r e m a i n d e r

and

set

sets

HTe m G+le

of

generating

together

is f i n i t e l y

subgroups

generated

groups

of our c h o s e n

H e _n G~I,

and e a c h

vertex

generating v

is f i n i t e

Y

a finite

by the

vertex

GROUPS

is f i n i t e l y

finitely

set g i v e n

FUNDAMENTAL

The

3.6

a vertex

of

H X,

of

G

then

H

lies

on the tree

X

so by

1.8.1,

group.

acts

g~,

g ~ G,

v E V(Y).

(Karrass-Pietrowski-Solitar

[73]).

If the

vertex

G

a free

are all

finite

subgroup

of finite

Proof.

Let

common

multiple

have

group

vertex

subgroup

say

Then

as desired.

THEOREM

groups

is a finite

of some

finite

stabilizes

H ~ gGv~1

H

A

and of b o u n d e d

then

has

index.

be a f i n i t e of the

a homomorphism

order

set w h o s e

cardinal

is the

orders

of the

vertex

groups.

G ÷ Sym A

whose

kernel

does

lowest

By 2.6,

not meet

any

we

47

THE FAITHFUL

conjugate has

of any v e r t e x group,

finite

index.

§3

CASE

so is free by 1.3,

Q

One of the m a i n results

we shall be proving

conversely,

every group with a free subgroup

be e x p r e s s e d

as the

vertex

groups

fundamental

3.7

index can

graph whose

order.

generalization

(Serre

If

[77]).

are finite t h e n for any free

rank F - 1 ........(Gi'F) ..

F

of finite

of the Schreier

index

1.2.

THEOREM

Proof.

is that,

group of a c o n n e c t e d

are finite of bounded

Let us note the f o l l o w i n g formula,

and c l e a r l y

Since

F

is i s o m o r p h i c

the n u m b e r

:

subgroup

l E(Y)

of

F\X

IF\G/Gel

_

subgroup

H

G.

structure

theorem

of

[

group of

index

in

G,

1

I%1

of any vertex group, by

F\X

1.1.

But

!G:r)

[

IGel

'

E(Y)

seen s t a t e m e n t s of

of finite

V(Y)

=

for vertices.

subgroups

and the vertex groups

is

Now

gives the desired result.

We have

F

1 [Gel

to the f u n d a m e n t a l

E(Y)

and s i m i l a r l y

is finite

does not meet any conjugate

of edges

[

Y

More G.

group of a certain

F

=

IE(FXX) I-IV(FhX)I+

D

about

free subgroups

generally,

Since

gives

rank

H

we can c o n s i d e r an a r b i t r a r y

acts on the tree

a description

connected

and finite

of

H

graph of groups

X,

the

as the f u n d a m e n t a l H\X +

Groups.

Here

1

48

II

FUNDAMENTAL

the vertex certain

groups

and edge groups

conjugates

of vertex

This d e s c r i p t i o n

actually

Kurosh

H.Neumann

[37] and

implies

3.8

these

results.

DEFINITION.

a tree

If

of groups.

denoted

simply

a colimit,

ease where

G

Y

~(~,T)

THEOREM

e E E(Y).

the subgroup

recall

T = Y

of

of

of

one case that

some terminology.

~:Y ÷

theoretic language then

with

theorems

G~o~ps

and the fundamental

are trivial

H

of edge groups

Let us mention

In category

~(Y,T)

and images

is a tree then

is not a tree, =

intersections

is called

group

language,

is

~(~)

is

is a ~ree product.

~(~)

is the co~roduct

groups,

H G . Even in the V(Y) v if all the edge groups are trivial

H ~(~IT).

(Bass-Serre,

Serre

[77]).

intersection

~et

H

be a subsroup

with each conjugate

of each

9f G -1, e

Then

H

=

where

F

~raph

H\X,

H\G/G v •

Y

of the vertex

that has trivial

cgrtain

[48].

and in group t h e o r e t i c

(or free produc~)

3.9

groups

generalizes

Here

If all the edge groups

then

are

We first

~(~).

GROUPS

F ~

H ( ~ (Hng~Gvg)) V(Y) H \ G / G v

is a free group,

and for each vertex

complete H

namely

set of double

the fundamenta ! group v

of

Y,

g

ranges

cose t r e p r e s e n t a t i y e s

of

of the

over a

~.

49

COPRODUCTS COPRODUCTS

4.

We shall n o w prove groups

4.1

edge

PREPARATORY T

some c l a s s i c a l

from the v i e w p o i n t

with t r i v i a l

of

§4

with

quotient

identifying

e

REMARKS.

More

Let

thus,

of

of groups

T :

be a tree and ~

~

e,f

two edges

We form the

with

and

f

graph

components

generally,

of if

Te

with

T - {e,f} S

repetition

of the p r e c e d i n g edge

made

is an e q u i v a l e n c e (e,f)

construction

set

E(T)/S.

T

by

It is e a s y to see up using the three

t o g e t h e r w i t h one more

for each pair

having

Tf.

is again a tree,

~e = If

T/S

of trees

with one less edge and vertex than

such that

graph

groups

T/(e=f)

that this quotient connected

of f u n d a m e n t a l

on c o p r o d u c t s

groups.

le = If,

graph

theorems

relation

• S,

on

edge. E(T)

then by t r a n s f i n i t e

we can obtain the quotient

It follows that

T/S

is

again a tree. We note that

any a u t o m o r p h i s m

an a u t o m o r p h i s m Of course,

of

4.2

THEOREM

families

statements

throughout.

(Higgins

of groups

[66]).

= ~ K i. Suppose ~:G ~ K I (Gi)e = K i for each i e I.

(Hi)~

=

Ha Ki

=

that r e s p e c t s

S

induces

K

there

for each

Let

if

~

(Gil

(Kil

and

I,

i e I),

and write

i E I)

G

=

Then for any subgroup

H

is a s u r j e e t i v e

exists

i • I.

r e m a i n valid

T

D

indexed by a set

K

such that

T

T/S.

all t h e s e

are i n t e r c h a n g e d

of

an e x p r e s s i o n

be two

H Gi, I h o m o m o r p h i s m with

H

=

2~

E Hi I

G with

50

II

FUNDAMENTAL

Proof

(simplified

version

cf [76']).

Clearly

T

with vertex

be a t r e e

specified There

element

we m a y

of

Let act,

edge groups, X : F(~),

There

induced

by

initial

be the

Z

as

a star where

vertex

one

of e a c h edge.

Group8

~,J~" :T ÷

standard

trees

t h e y act

Z

Let

with

K : ~(~C).

surjective of

[77],

is n o n e m p t y .

for e x a m p l e ,

in p a r t i c u l a r ,

can v i e w

I

of g r o u p s

a n d we t h i n k

one

in C h i s w e l l

that

G : ~(~),

is a n a t u r a l

e,

(In fact,

trees

Z : F(~)

respectively;

sets.

I;

is the

and

given

assume

set

I

are t h e n n a t u r a l

trivial

of t h a t

GROUPS

freely

on the

graph morphism

X + Z

as a q u o t i e n t

NXX,

on W h i c h

where

N

G,K edge

g r a p h of

X.

is the k e r n e l

of

e.) Let

H

act on

and act on

Z

we m a y v i e w quotient Among

edge

terminal

of

quotient

Y.

H\Y

Suppose

that

transversal on V(S)

E(Y)

those

S

the

= V(T)

in

÷

the

inclusion map

composite of

of

for w h i c h

Y = X.

H\Z

:

X H

H + G,

H + G + K.

X,

more

that

lie

acts

By Z o r n ' s a natural

K\Z

=

Thus

precisely,

a

c o p y of Y

T

in

for the theorem

Finally,

Z

Z,

f r e e l y on the

Lemma there

exists

a

surjection

We can t h e n b a c k to a c o n n e c t e d

H-action. implies

for e a c h

above

T.

is an i s o m o r p h i s m .

structure

= I.

Y

We t h e n h a v e

this

lift the c a n o n i c a l

the

H-graph

H-trees

for e x a m p l e , such

along

X.

consider

set;

along

as a q u o t i e n t

H-tree

X + Y ~ Z,

by p u l l b a c k

by p u l l b a c k Z

the

X

Because H

=

H

acts

freely

H H i with V(S) i e I, (Hi)~ = K i since

51

§4

COPROOUCTS

H. i

is the Thus

that

H-stabilizer

it s u f f i c e s

H~Y + T

injective mapped

to d e r i v e

is not

on edges.

to the

same

image

in

edge

Replaeing

y

y'

are m a p p e d

el en Yl ' ' ' ' ' Y n Clearly

Z,

n m 2;

be some

but

lying

the d e s i r e d

element

h

H.

then

The

on

E(Z).

p'

=

h y ~ l '''" ,hy[i,Yi~le" ' ' ' ' ' Y ne n "

lie

in d i f f e r e n t

but

as

is r e d u c e d gives

is not

(That

H-orbits

of e d g e s

and h a v i n g

common

different

H-orbits.

Now

2. of

initial

so

equivalence

relation

on

hy,

y'

I h e H, for

Yi+l

Yi+l

so

h K

acts

a n e w path

= hy 1 , and hz

of

for

since

is a p a i r

in y'

= z

Y, = Yn

in

Z,

for w h i c h

n

way e v e n t u a l l y

common

vertex,

and

of e d g e s

:

image

in this

"

is a tree

Yi

Z

and

Z,

Z

If

hy

having

are

=

= y'

in

and

hy i

common

or t e r m i n a l

E(Y);

P

y

P

Yn

is a p a i r

Here

y,y',

{ (hy,hy'),(hy',hy),(e,e)

of

Consider

Continuing Y,

Y'

all of

= 1.)

and h a v e

a geodesic

by at least

a pair

ha

that

assume

hz i = zi+ 1 = z i

so s t a b i l i z e s is,

=

image

Otherwise,

freely

H\Y

Let

= z~ei

This m e a n s Z

say. Yl

yi,Yi+ 1

is not

H-orbits.

z I = z = zn

an e d g e

supposition

and h a v i n g

we m a y

z

having

stabilizes

P'

of

vertex,

~ei+l -i+1

property.

of

a common

Z,

edge

with

H-orbits

e E(Y).

of Y

of

y,y'

H-mutiple,

n = 2.

i

f r o m the

in d i f f e r e n t

in

Z.

or e q u i v a l e n t l y ,

where

same e d g e

of

are two e d g e s

having

a repeated

in d i f f e r e n t

some

Y

we want

so there

with

T,

of

to the

has

Y

Hy'

by a suitable

z~l ,...,z ~ n,

are

of

vertex

a contradiction

Hy,

be a g e o d e s i c

must

correct

an i s o m o r p h i s m , Say

We s e e k t w o e d g e s common

of the

but

e e E(Y)

it is c l e a r l y

}

image

in

lying

in

Z

is an

reflexive

and

52

II

FUNDAMENTAL

symmetric,

and

different

it is v a c u o u s l y

H-orbits

the r e s u l t i n g

and h a v e

quotient

quotient

H-tree

see t h a t

H

of

acts

contradiction

to the

properties.

D

4.3

COROLLARY

there

for e a c h

of

Y

above

that

and

8:F + K

Y

in fact

Further,

so we h a v e is t e r m i n a l

F

a

it is e a s y to

obtained

a

with these

be a free

K = HK i I homomorphism.

be a s u r j e c t i v e F = H Fi

lie in

By 4.1

is a t r e e ,

Z.

Let

an e x p r e s s i o n

y,y'

H-stabilizers.

E(Y'),

[57]).

exists

Let

X

x E X

choose

lying

in some

the p a i r s

be a free

K i - {1}.

(x,m), Then there

to

Yx,1...Yx,nx ,

i e I

let

Gi

Yx,m

for w h i c h

group,

such that

and

each

Yx,m

to

f r o m 4.2.

D

(cardinal)

x E X.

subgroup lies

e

Recall

of

Y = {Yx,m }

a n d we v i e w

F~ = K.

restriction

Let

kx, m

of e l e m e n t s

in

F

F,

(Fi)8

of

= Ki

F

coincides

of a g r o u p

so

= Ki

by

with

is the

of

For e a c h by t h o s e

~ Ki

for e a c h

Since

the

the r e s u l t

it.

x E X

is a h o m o m o r p h i s m

smallest

to g e n e r a t e

G.

by

group

each

generated

3.3. 8,

free

sending

(Gi)e

kx, m

indexed

be the

Then there

kx, m

(Gi)~

required

F ÷ G

freely

K i.

to

G

for e a c h

each

be a set

Let

G

and

with

as a s u b g r o u p

Thus

t h a t the r a n k

set for

: kx,1...kx,nx

is a m o n o m o r p h i s m

be the

sending

x8

1 ~ m s nx,

Y.

e:G + K

generating

an e x p r e s s i o n

on

follows

fact

because

i • I.

Proof.

i ~ I,

on

(Wagner

be a e o p r o d u e t , Then

Y'

lying

freely

transitive

trivial

graph

Y

GROUPS

number

now

53 COPRODUCTS

4.4

THEOREM

rank(A HB)

(Grushko =

rank A

Proof.

Clearly

the

group having

free

surjective with

[40],

rank(AH the

homomorphism

(FA)8

r a n k ( A lIB)

= A,

=

(FB)B

rank

F

B.H.Neumann

+

rank

B)

~

same

[43]).

For any s r o u p s A,B,

B.

rank A r a n k as

8:F + A H B . = B.

§4

+

rank

A ~ B, By

rank

FA

>-

rank A

+

4.3,

+

rank rank

Let

so t h e r e

Hence

:

B.

FB B.

F =

F

is a

FA ~ F B

be

CHAPTER

III

DECOMPOSITIONS

This

chapter

derivations

and g r o u p s

by a t h e o r e m theory,

group

gives

permits

will

rise

quite

G

= G e

a detailed the

)

analysis

on w h i c h given

of the

derivation. to t r e e s

into the Bass

on a f i n i t e l y

is the

generated

the g r o u p

by D u n w o o d y

group

The

and h o w

concept

acts. [79],

it

intermediate

of

of

G,

{ e l qe = 1}

subtree q~ n G e

such t h a t

T

of

link that

of an a l m o s t

Then

Y.

G

Y,

and a f a m i l y

For e a c h

we h a v e

graph

edge

a connected

a family

(qe 1 e e E(Y)

is the e d g e e

of

graph

Y

)

set of a we

set

of g r o u p s

Te

le, Te:~(e)

÷ ~(le),

is t h e n

isomorphism,

for a c o n n e c t e d

of s u b g r o u p s

defined

a

a tree,

set

exemplified

A GROUP

Groups

There

when

any d e r i v a t i o n

we are g i v e n ,

le

~:Y +

which,

a relationship

be a group.

of e l e m e n t s

G

[68]

of such

derivations

v e V(Y)

maximal

on trees,

between

set.

Suppose (Gvl

relationship

in some way to a tree

DECOMPOSING

Let

curious

acting

that

to a d m i t

connect

invariant

I.

says

description

decomposes

the

of S t a l l i n g s

Se~re

An a c t u a l

explores

by

~(y)

~(Te)

a natural

map

= Gy

given

of G.

by

~(~,T)

t h e n we call the

Y-decomposition

(y ~ Y) w i t h

The

the maps

g ~ g,gqe ÷ G.

system Gv,G e

respectively.

If this

is an

(Gv,qe I v e V(Y), are

called

e ~ E(Y)

the v e r l e x

)

groups

56

Iii

DECOMPOSITIONS

and edge

groups

interest

in any d e c o m p o s i t i o n ,

of

of the

"a d e c o m p o s i t i o n

structure

of

Y

decomposition;

(Gvl

and the

For e x a m p l e

{G}

they

and we

v e V )"

are

shall

of the

is a d e c o m p o s i t i o n

of

d a t a of

usually

and t a c i t l y

specification

the

speak

assume qe

G,

that

loosely the

are u n d e r s t o o d .

called

the t r i v i a l

decomposition. For a n o t h e r

example,

II.4.2

is r e a l l y

a statement

about

decompositions. Associated tree

on w h i c h

on a tree in

with

X

(gvl

G

X

acts,

v e V<S)

]

is a c o n n e c t e d

groups

that

results

transversal

To state notation. denote

G,

resulting

v e V(S)

and t h e n

for

We then have

vertex

v

for

of Gv

S

S

a family

G-action, S'

acts

choose

e e E(S) we m a y

) a

a

). choose

to be one of the v e r t e x

decomposition. allows

us to w o r k

with

trees

to o b t a i n

decompositions.

the next

pairs,

= g@hx

G

transversal

S' = (gvVl

( g v G v g ~ , q e I v e V(S),

we can a r r a n g e

the i n d u c e d

preserves There

of

)

suppose

an a r b i t r a r y

so that

for the

result

If a s u b g r o u p

of o r d e r e d gh@x

is,

G

(qe I e e E(S)

is a w e l l - d e f i n e d

Conversely,

of

correspondence about

there

of e l e m e n t s

for any s p e c i f i e d

of the

This

1.5.3.

G

T h e n we can c h o o s e

G\X-decomposition

gv = 1,

of

G-action.

family

Moreover,

by

and that we are g i v e n

for the

connecting

any d e c o m p o s i t i o n

G-set g@x,

for all

stabilizers

are two ways

H

of G

whose g c G,

h e H. and

it is c o n v e n i e n t acts

elements x e X,

Here

induces

of c o n s t r u c t i n g

to h a v e

on a set are

with

the m a p

X,

let

equivalence defining

G@HX

classes

relations

X ÷ G @ H X , x ~ l@x,

a bijeetion G@HX

some m o r e

H\X + G\(G@HX).

directly

-

either

57

DECOMPOSING

choose

a transversal

S

in

G-set

V G/Hs, or c h o o s e S H - a c t i o n and m a k e S x X where

gs : s'h,

1.1

THEOREM.

the

G-action

Gv-tre___~e there

Let

X

a left

be a

there

G-tree

X

Proof.

Define

where

:

To d e f i n e

the

of

G-set

for the

g(s,x)

=

right

(s',hx)

a transversal

is g i v e n e

G

the

for e a c h X,

Ge

of the

in

X

for

v • V(S)

is f i n i t e G-forest

then

v G@GvXv, V(S)

E(X).

G-sets

V V(S)

E(X)0

in

and t a k e

x • X.

S

consisting

w i t h a c o p y of the

S

h e H,

G-tree,

If for e a c h e d g e a

H-action

G - s e t by

s,s' • S,

tosether

v(~)

for the

a transversal

and s u p p o s e

X v.

exists

g e G,

X

§I

A GROUP

G@ G V(X v) v

and

is a c o p y

of

incidence

maps

E(X)

E(X) on

=

with g@e

E(X)0

E(X)

v

V V(S)

G@ G E ( X v ) v

4-~ E(X)0

• G@ G E(X v)

e 4-+ e0.

set

V

l(g®e)

= g®le

• G@ G V ( X v ) ,

and

similarly

for

T.

Thus

V

contains

the

everything Every g ~ G,

G-forest

using

element

V G®GvX v a n d it r e m a i n s V(S) the e d g e s of E(X)o. of

e E E(S).

E(X)0

Observe

such that

le : g e V e .

a subgroup

of

Gv

;

can be w r i t t e n that

there

in the

exists

a

to l i n k up

form

ge E G

ge0, and

v e • V(S)

Then

G -i ~ G l ( ~ e ) , t h a t is, is g~e G e g e ge e as it is a f i n i t e s u b g r o u p it s t a b i l i z e s a

e

vertex

of

Xv

,

by

1.8.1.

Choose

such a vertex

and d e n o t e

it

e by the l(ge0) g

formal =

symbol

gge@'~le0

stabilizes

eo

,gele0 -i ,

Now define,

' • G@ G X v ; ve e

then

gge

= ge h

this

for e a c h

g ~ G,

is w e l l - d e f i n e d ,

where

h

lies

in

for if

Gv

and e

stabilizes

'£l~eo'

We d e f i n e

T

on

E(X)o

similarly.

Thus

58

III

DECOMPOSITIONS

is a w e l l - d e f i n e d incidence remains ÷ X

maps

g ~ G,

that

v ~ V(S),

and

le0

containing

E(X)0

bundle

that

no r e p e a t e d

so lies

in a fibre,

of

X

if

is

vertex

X

So

of

THEOREM G

$iven

with

sends

each

each have

e,

It

vertex

gv,

e E E(X).

the

X

morphism,

X ÷ X.

Here

The p o i n t

is c o l l a p s e d

under

is i m p o s s i b l e ,

images

in

fibre

then

X

to e a c h

that

[73]).

v e V )

edse

sroups, (Gwl

and

w • Wv )

Gv,

all edge

G.

D

finite

then

so does

a

is c o n n e c t e d , is O

following.

v e V

then

w • V W v ) of G, for some V If m o r e o v e r t h e d e c o m p o s i t i o n s

groups

have

is a d e c o m p o s i t i o n

if for e a c h of

of c o m p o n e n t s

which

the

in

no c i r c u i t s .

is a tree.

we o b t a i n

(Gvl

has

vertex

to d e c o m p o s i t i o n s

image

to a v e r t e x

components

X

a

Any c i r c u i t

X

other,

Hence

inverse

some p a i r

of the

over

is c o n n e c t e d .

If

the

X + X

so

is

so we have

is a tree.

(gwGwg~]

of

of the

the map

to the

graph

any fibre,

are c o n n e c t e d

gw °f-~ G.

decomposition

G-graph.

Consider

not c o n n e c t e d ,

a decomposition

elements

is a

g@G Xv v e0 to

a tree,

which

this

finite

over

edges

(Cohen

decomposition

definitions

g@GXv for w h i c h gv : le v to the s m a l l e s t s u b g r a p h of

X ÷ X

But as the

the t w o c o m p o n e n t s

1.2

tree

is

in common.

Translating

each

is,

such that

impossible.

X

is a s u r j e c t i v e

with

Similarly,

that

is a tree.

of

of t r e e s

of any v e r t e x ,

and f r o m the

to be in the tree

the r e s t r i c t i o n

fibre

~

collapses

of c h o o s i n g that

it is c l e a r

to c h e c k that

graph,

there

there

is

exists

choice of the

the r e s u l t i n ~

of Gv

a

59 DECOMPOSING 1.3

REMARKS.

Suppose

stabilizers.

Then

XI,

X2

for each

A GROUP

are

vertex

§I

G-trees vl

of

with

XI,

finite

G

edge

acts

on

X2,

Vl

so by 1.1,

we o b t a i n

over

Now for any

XI.

a new

G-tree

subgroups

difficult

to v e r i f y

that

(1)

G/H x G/K

=

there

XI

which

H, K

are

v HgKeH\G/K

of

is a fibre G,

isomorphisms

G / ( H n g K g -*)

bundle

it is not of

=

G-sets

G@H(G/K).

^

Hence,

from the

V(X~)

=

V(X~)

construction × V(X2),

of

X,,

E(X,)

=

we

see that

E(X,)

as

v (V(Xl)

G-sets

x E(X2)

).

A

By s y m m e t r y , X2,

we h a v e

and as

E(Xz)

=

G-sets,

E(X2)

G-sets.

v (V(X2)

v (V(X,)

THEOREM.

1.4

V(X I) = V(X2)

Proof. for

If

and

Let

~:VI

E(~):~[E1]

÷ ~[E2],

as usual,

for

ele I + ...

+ enen,

v

to

and that For any Gel

E(~),

It is e a s y

el E El,

stabilizes

the

E(XI)

,e~n

Gel

in

X2

as

such

D

that

as

G-sets.

and s i m i l a r l y

the a d d i t i v e

(Tel)~),

map

el e El.

X2(v,w) is the

).

(Here,

for

geodesic

E(~)

is

in

X2

~[G]-linear,

inverse.

= G le I n GTe i ~

geodesic

× E(X,)

= E(X2)

to show that

are m u t u a l l y

= V(X2)

in this

El = E(XI),

(tel)a,

e11,.. . e

that

G-trees

we are w r i t i n g

where

over

is,

and c o n s i d e r

el ~ X2(

E ( & I)

says

bundle

and

V(il)

v (V(X2)

VI = V(XI),

V2

Thus

are

then

~ V2,

v,w~

w.)

E(X2)

X2

G-sets,

Let us w r i t e

X2.

from

X,

~

).

that

is a fibre

x V(XI),

result

= E(X2) ; )

which

V(X2)

next

x E(X2)

as

=

x E(XI)

the

E(XI)

X2

G-tree

V(X2)

Curiously,

circumstance, E(XI)

a

G (lel)~

from

n G( Tel)~'

(lel)e

to

so

(Tel)~.

60

III

DECOMPOSITIONS

Let

H

be any subgroup

of

El

fixed by

E,

whose

carries

E~

to

E(e):~[E~

~

~[N]-modules,

Hence

there

~[E,(H)]

rank

is induced

is p r e c i s e l y

these

numbers

subgroup

H

vertex

vl

V(XI)

v~

v E(X2)

Proof. X 2.

of

a

that

XI

XI,

G

X2~

Gv2

V(X2)

Let us write The hypotheses

morphisms

~:V;

E(~,8):~[V,

is,

v E2] +

E,

X2

of

H

in

isomorphic E2

to

as

G-sets.

Lemma for

G-trees

such that

a vertex

of

X,.

(le 2- Te 2) + XI(

El = E(XI),

and for each Then

and similarly

to the existence There

defined c

(Te2)S,

to show that

~ [G ]-linear maps, theorem

for each

G-sets.

8:V2 ÷ V,.

~

[]

G-trees.

stabilizes

e2

Since

no m a t t e r which

X2,

vl

But this G/H.

of

vle + X1(vl,vl~8),

G.

= ~@~[N]~[Ez(H)]

a vertex

as

to

on the quotients,

stabilizes

v E(XI)

~[E~]

the same rank.

and

be

E(e)

are isomorphisms

isomorphism

of Schanuel's

~

inverse

and these

we see El = E2

~ [ V 2 v E,]

proof of the p r e c e d i n g

carries

of

and similarly,

of orbits

vl

mutually

E ( ~ I)

they have

are equivalent

It is s t r a i g h t f o r w a r d

shown that

~@~[N]~[EI(H)]

Vl = V(X,),

÷ V2,

We have

~[N]-linear

and

v1

for the set of elements

is the n o r m a l i z e r

the n u m b e r

Let

=

H.

Therefore

--

vertex

N

an analogue

of

for the set of elements

- E~(H)];

is considered,

THEOREM.

El(H)

~[E~],

are the same for

We also have

1.5

where

groups;

~

~[E~

= ~[E2(H)].

as abelian

E~

and similarly,

E(~):~[E~]

- El(H)]

Write

is precisely

~[E~],

Hence

G.

and write

stabilizer

~[E~].

of

H,

of

shows

of

for

G-set

is then an additive

map

by

VI;

and

(Ie2)8), E(e,8)

and

e 2 e E2. E(8,e)

and the argument that

V, v E2

=

are used in the V2 v El.

D

61 OECOMPOSING

It is interesting hand

to note that

side of the equation

§1

A GROUP

it is now clear why the right

in 11.3.7

does not depend

on the choice

of the tree. Let

X

edge

e

G1e,

G e

be a of

X

G-tree,

X',

with

V(X')

then,

that

stabilizers, geodesie

for any r e d u c e d

in the same orbit,

1.6

G

V1

Thus there If

X~

injeetive

Vl

stabilizes

with image

then

eS,

orbits,

we obtain

number

= E(X) of such

with

G v = Gw v

to

X

one of

a new

- Ge.

If

steps,

we

V

of

in the

w,

V,

any since

G-trees

with

be the set of vertices of

X2;

G-sets

and

V, = V(X,)

V2,

so

8

then,

all the

so

Vi

for, v0,v I

finite

v 2 , . . . , v n.

V(X),

image

be two

of

edge

being

and so on for

a vertex

finite

is reduced,

stabilizers,

Vl = V(XI), 8~

X

the same stabilizer;

and has

morphisms

is reduced

if

for some

say for instance

E(X')

from

and

XI, X2

Let

Similarly, reduced

Let

exist

Te"

if

X

G-subset

is injective

stabilizers. that

in

and t h e i r

for any

REMARKS.

to

G-tree

and have

must be equal;

e:V ÷ V(X)

in d i f f e r e n t

- G1e,

v , w e V(X),

n G v n ~ Gel ~ Gvl,

particular,

e

if,

G-tree.

for any

conjugate,

Te

In this event,

= V(X)

v0,e~l,.. ~e~n,v n

= Gvo

is u n r e d u c e d

after a finite

lie in the same orbit, Gv0

X

and

the other.

at a redueed

Notice

~e

then by " c o n t r a c t i n g

is finite,

arrive

We say

that has

contains

G1e = GTe,

G\X

G-tree.

lie

and In

G-set m o r p h i s m G v c_ Gve for veV. finite

edge

vl

of

XI

and define

V2

analogously.

~:Vl then

+ V(X2),

such

8:V2 ÷ V(XI).

Be:V2 + V(X2)

is

is injective. V2 = V(X2),

are automorphisms,

so

and

Xl,

X2 are

V(X I) = V(X 2)

as

82 llI

OECOMPOSITIONS

G-sets,

2.

and by 1.4,

= E(X2).

D

CUTS

Let

G

be a group and

Cayley graph

2.1

S

~A

For any graph of

A

in

X

that have to be removed

~A

=

{e ~ E(X) I e

We denote

by

V6A

is c o n n e c t e d

is d e f i n e d

lying

in

of

6A.

A

then

lies

6A

disconnected

if and only if

~B e 6A,

A cut of

many

that

and then

X if

X

edge of

subgraph F

of

lies in

subset F AS

A

A

We make the c o n v e n t i o n

Clearly

X,

of

6A

so,

X

that if and of

X.

union of the

6A

A

subgraph of

B

is such

A.

is finite.

then a cut has only finitely is again a cut.

of

G

we denote set

S-coboundary

D

by

A,

of

we shall write

that the empty

Notice that

A}.

in particular,

such that

of

is,

component

AS that

if and only if both vertices in the

the

that

is empty

in a d i f f e r e n t

A,

X

and one not in

vertices

which has vertex

is no risk of c o n f u s i o n

S-coboundary.

A

each of w h i c h

We shall be i n t e r e s t e d there

for the

of

is a union of components

is c o n n e c t e d

For any n o n e m p t y

A

from

has a (nonempty)

is a subgraph

components,

largest

of

A

B

A

is the disjoint

of the c o m p o n e n t s

Notice

r

to be the set of edges

to d i s c o n n e c t

coboundaries

that

Write

G.

and subgraph

has one vertex in

only if each component X

X

the set of all those

belong to some edge

If

a subset of

F(G,S).

DEFINITION.

coboundary X

E(X~)

A, 6A

subset of

for any subset

A

of

the is,

an

lie in

A.

6sA = 6(As) ; as in place

G

of

has empty G,

the

~S A-

63

CUTS complement If

6A

S-cut

then

A A

invariant

If

In t h i s

event,

and

if

s e S

S

S-cut

are

infinite,

AS

of

A

then

A

G

of

and

G

A

is

A

To

f r o m 1.§8.

if in

B,

is a l m o s t

is c a l l e d

and

e q u a l to

almgst-risht-

0

G

is an

As = A

generates

G,

A.

A is c a l l e d

and

contained

B ~ A

g ~ G.

G;

terminology

A,B

of

of

as

is c o n n e c t e d .

is a l m o s t

for e v e r y

for e v e r y

S-coboundary

an

following

A

A subset

same

or if

A subset

Ag 9 A

PROPOSITION. As 9 A

A*

subsets

A ~ B

A = B.

if

the

t h e n we say

a

denoted

and

if it is e m p t y

For any

A ~ B.

h a s the

is c a l l e d

DEFINITION.

write

if

G

we r e c a l l

is f i n i t e

2.3

in

S-cuts

A - B

B,

A

if b o t h

connected

describe

2.2

of

is f i n i t e

a proper called

A*

§2

S-cut

for a l m o s t

if and o n l y every

s e S.

then

A

is a l m o s t - r i $ h t -

fact t h a t

A

is an

invariant.

Proof. only

This

if t h e r e

as ~ A

or

is c l e a r

are o n l y f i n i t e l y

as ~ ~ A.

The r e m a i n d e r S-cuts

2.4

needed

LEMMA

f r o m the

(a,s) e A x S

if and

with either

Q

of t h i s

section

in the n e x t

(Cohen

many

S-cut

[70]).

is d e v o t e d

to r e s u l t s

about

section.

If

S

generates

G

then

for any

a

S-cuts

Proof.

A,B

of

G,

A

=

{ a E A I aB ~ A

C l e a r l y we m a y a s s u m e

that

A

or

aB* ~ A } .

is n o n e m p t y ,

and h e r e

64

III

DECOMPOSITIONS

{gEGI gB~A} =

{g~

GI

6(A*

ngB)

¢}

=

(2) {g E G I A* n V6gB =

Since

A

Then

{g e G I V6gB m A

is an

connected

=

S-cut

subgraph A

~

¢

and

=

V6A* n gB}

V6A~gB*}.

and

F

is c o n n e c t e d

~--A of

F

containing

{a E A 1 aV6B ~ A }

since and

{ a e A[

aV6B ~ A

{a e A I V 6 a B ~ A

A

{a e A 1 V6aB { A {a c A I aB ~ A

2.5

DEFINITION.

set of all

of

G)

setting

A ~ B

(A,B e P(G))

elements

of

that moves

carries

A

that

is,

~

formulation most and

B;

A ~ B

Since

only

shall if

In p a r t i c u l a r ,

for e a c h

subset

[A]

=

in

aB S

lies

in

by

(2).

P(G) relation

many

~-A

~B are

finite

or

and

aBe}

aB

or

IA - B I

aB*}

D

(that

is,

defined

elements

the by

of the of

G

and

and t r a n s i t i v e ,

be m o r e

IA - B 1

if

since

is c o n n e c t e d

is r e f l e x i v e

We

under

¢}

is a p e r m u t a t i o n

finitely ~

=

G,

be the

if t h e r e

if and o n l y

it is a p r e o d e r ,

and

~

clearly

are c o m p a r a b l e

V~A

set of

let

is a p r e o r d e r .

IB - A I. B

P(G),

into

6A

aB* ~ A }

On the p o w e r

subsets

G

and or

is f i n i t e

6--A lies

since

a finite

is a l m o s t - r i g h t - i n v a r i a n t

~-~n a6B

and

choose

6A,

V6B

and

we m a y

interested

is f i n i t e is f i n i t e

and

in the is at

then

A

~ .

~

induces A

of

{ B¢G

G

I

an e q u i v a l e n c e we have

A~

relation

an e q u i v a l e n c e

B .4, A } .

on class

65

CUTS

Let

P(G)

denote

equivalence will

be d e n o t e d

there

similarly

for the

involution

LEMMA.

S-cuts

of

Since

right

*:P(G)

(ii)

{ g e G I g[B]

= [A]}

Let I = A

this

by 2.3, = A*},

also h o l d s

{ g • G I gB = A

or

@

{a • A I a B c A

A

~

{a e A I a[B]

or < [A]

in place

comparable

{g • G I g[B]

DEFINITION.

S-cuts setting

of

G, A

~ B

B*

on

and

by

P(G)

on

~(G)

not

concern

reverses

G

~

(and us).

so induces

0

let

under

Then

A,B

~

be p r o p e r

A.

of or

< [A]

or

Let

Ps(G)

But we also

in place

A

b E B,

and t h e r e f o r e of

C

B,

Cb ~ A

have

must

be finite.

we d e d u c e

that

So by 2.4

so a f o r t i o r i < [A]}.

The c o r r e s p o n d i n g

gives g[B*]

> [A]}.

Since

g[B],

g[B*]

we deduce g[B*]

< [A]},

denote

a partial

(A,B c Ps(G))

for any

C~A.

a[B*]

~

and d e f i n e

< [A]}

is finite.

or

{g ~ A*I g[B] > [A]

2.7

does

P(G)

is finite.

aB* c A},

A~

~

A*

With

with

A

on G

P(G).

g[B*]

a . C ~ A ~=

so

result

are not

or

and thus

gB* = A}

A

9

on

C = {g e G I gB = A}.

a

C = {g e G I gB* Since

this

such

Then < [A]

C~A~

of

G-action

generates

{g • G I g[B]

so

action

but

*,

of all

induced

A ~ A*,

involution,

S

order

a left

action,

(i)

Proof.

consisting

left

÷ P(G),

Suppose G.

the

is i n d u c e d

an o r d e r - r e v e r s i n g

set

The p a r t i a l

~.

~

2.6

quotient

classes.

respects

The

the

§2

the

order

if e i t h e r

and h e n c e

(i),

set of p r o p e r ~ 16AI

on >

Ps(G) 16B I

or

(ii).

connected by Am B

0

III

and

DECOMPOSITIONS

I~AI

= I6BI .

if for every

(3)

A subset

g • G

A ngA,

and

A ~ B.

of

Ps(G)

will

be called

full

at least one of the four sets

A ngA*,

components

A* n g A *

of the form

xB,

x • G,

B • ~,

0

The f o l l o w i n g and Cohen

2.8

A• ~

A* n g A ,

has all its infinite

#

any finite

(Dunwoody

[79]).

f a m i l y of p r o p e r

in a finite

G

The statement the left

Suppose

connected

S

of Bergman

generates

S-cuts of

of this argument

to be a p r o p e r

connected

of the t h e o r e m

G-action,

suffices

results

[68]

G

G.

Then

is c o n t a i n e d

full family.

For the p u r p o s e s

S-cut of

and simplifies

[70].

THEOREM

Proof.

generalizes

show that any thin

S-cut that

concerns

orbits

and as each orbit

to c o n s i d e r thin

S-cuts,

S-eut belongs

let us define

in

thin

and in fact to a finite

contains Ps(G)

contains

a thin 1.

under S-cuts

it suffices

it to

family of thin

S-cuts. Let us now a s s o c i a t e family

~(A)

of thin

with each thin S-cuts

least one of the four sets the

form

xB,

x • B,

~ A

S-cut

such that

A

of

for each

(3) has all its infinite

B • ~(A).

We c o n s t r u c t

G

a finite geG

at

components

such an

~(A)

of

as

follows. We are the sets

concerned

only with those

(3) are infinite,

bad elements

of

G

and there

since by 2.4,

g •G

for which all four of

are only f i n i t e l y m a n y

such

67 CUTS G

@

bad

{ g • G I gA g

or

gA*

we n o w choose

'

is c o n t a i n e d

some thin

A n gA

(4)

summands (3),

to put

A*

in

}.

For each

$(A).

'

\/

~

A*

n

=

2 I 6(A n gA)

u 6(A ~ n gA)

:

2[

;

~A u ~gA

of

= 16AI.

call it B

choose

summands

gA*

has a

c e C

< 16AI

B be the intersection then it is a thin B ~ A. of

noting

If B

B

choose

that

thin

the chosen

so

~'c

(3) that

S-cut

a

8(A)

For any thin

S-cut

the

1.

then

for each

so

for each of the

set

component

noting

B

8(A).

that

case,

let

is connected Since

B cA,

infinite CIC

in

component Z(A),

clC ~ A. finitely

with the desired

of

intersections

latter If

it in

16AI,

call the elements

successors

In

8(A),

and put the thin

this process

A,

in

contains

and we put

c E C

one of the

c'IC

I

or all four

for each infinite

~ A.

is not connected,

a family

S-cut

and put

u 6(A* n gA*)

< I~AI,

case,

< 16AI;

16(cIC) I < 16BI s

Repeating we obtain

16BI

u 6(A n gA*)

in (4) is

In the former

B,

16(~1c) 1 ~ 16sl

C

or

l~(An gA) I + 16(A* n gA) I + 16(An gA*) I + IS(A* n gA*) 1

Thus one of the four

C

gA

A

n gA*

. n

in

S-cuts

~ A

\ A*

§2

many bad

properties.

of the constructed

g,

For each set

Z(A)

A. A

of

G,

consider

the

family

%

of thin

68

III

DECOMPOSITIONS

S-cuts

generated

clearly

a full

Suppose

~

finitely

many

that

is in

for some

n ~ N

In this

technical,

of

We

shall

here

elements; H.

set

such

(It is m o r e

Lemma

An .

the

implies

So in of

is e v e n t u a l l y of

Since of

n a N,

where

sequence

a ...

subset

this

lies

are

contradicting

~

it c o n t a i n s

~A

c_ ~A n .

a subgroup how certain

although in the

will

all

1

in

equal, An

Since

is

A n ~ An+ 1.

Thus

these

study

of

G.

conditions

conditions

appear

of d e r i v a t i o n s ,

be a p p l i e d

in the n e x t

give

rise

to

rather

and the

section

to give

a

of d e r i v a t i o n s .

say that

a generating

H

we e x a m i n e G;

deep u n d e r s t a n d i n g

has o n l y

By d e f i n i t i o n

~A

is finite.

A0,A1,A2,... of

is

D

and

they occur

proved

~

This

THEOREMS

be a g r o u p

decompositions

16A2I

J6An],

,

Tree

= naN n An .

such that

as d e s i r e d .

section

~

any f i n i t e

An = A

K3nig

Hence

A

But the

S-cut

successor

J6AII

Let

thin

sequence,

Nth term.

n a N.

this m e a n s

G

results

a

Also,

DECOMPOSITION

Let

the

an i n f i n i t e

successors.

to s h o w t h a t

successors,

....

is some

is f i n i t e

3.

it r e m a i n s each

say f r o m the

connected

chosen

Since

16AoI

A N ~ AN+I n

so t h e r e

and

taking

A 0 ~ A 1 ~ A 2 ~ ....

is n o n e m p t y .

~A n

under

is a c h o s e n

integers

constant,

A

~

An+ 1

particular,

means

family

chosen

n,

positive

A

is i n f i n i t e .

there

for e a c h

by

G

is f i n i t e l y

consisting a set w i l l usual

of

H

be said

to s p e a k

~enerated

and

finitely

to g e n e r a t e

of the p a i r

over

H

many G

(G,H)

if

G

has

other

finitely being

over

finitely

69

DECOMPOSITION generated, related

but we shall not be doing

to the cuts of the previous

consequence

3.1

Suppose

S

and its finitely

many

We shall

say that

each edge group decomposition is specified

from I.§8

H-subset

THEOREM

over right

H.

over

a subset

#

for each A n gB,

1

[79]).

G

A

of

H

Then

S-cut,

are asain D G

is finitary

if

graph

is finite.

A

H if there

is contained

is simply

in

of

G

is called H-subset

B

the additional

Suppose

G

an almost of

hypothesis

is finitely

ge G A* n gB,

almost

right

such that Gv-&Ubset

(5).

generated

set of a l m o s t - r i g h t - i n v a r i a n t

such that

is a finitary

(Gv I v • V),

G.

of this

and each

A,B • ¢,

A n gB*,

A* n gB*

one of the four sets

is finite. Then there

G

v H" a decomposition

form of the main result

in having

be a finite of

is an

G.

for some right

only

(Dunwoody

Let

H.

is said to be over

such that

at a p r e l i m i n a r y

H-subsets

(5)

is

vertex.

A ~ B

- differing

of

over

G

components

and the u n d e r l y i n g

vH • V

that

if

of

decomposition

a decomposition

We now arrive

3.2

concept

by th~ following

finitely

H-subset

(Gv I v • V),

a vertex

G

H-subsets

a given

G~

a distinguished

section

right

is finite

of

For example,

risht

S-connected

almost-ri$ht-invariant

right

This

section

generates

any a l m o s t - r i @ h t - i n v a r i a n t

Recall

so.)

of 2.3.

LEMMA.

with

§3

THEOREMS

decomposition

for each of

G.

A E ~,

of

G

over

an__jd v • V,

H, A

is an

70 III

OECOMPOSITIONS

Proof. of

Let

~

are

that

~

S

generate

S-cuts.

that

H-subsets),

involution

*.

Let

=

Z

G~,

A n B,

loss

over

the e l e m e n t s ~

{gA I g • G,

E,

one of the

A* n B,

A n B*,

so the e l e m e n t s

of

~

we m a y

assume

are p r o p e r

is c l o s e d

is,

A,Bc

H,

of g e n e r a l i t y

and that

that

for any

(6)

finitely

Without

is n o n e m p t y ,

(and r i g h t

G

under

A • ~}.

S-cuts

the

From

(5) we

see

four sets

A* n B*

is finite.

Let

E

=

{[A] e P ( G )

partially

ordered,

u n d e r the

actions

(We r e m a r k the n a t u r a l case

one

does

under Hence

~

of

G

if

(E, ~)

from

÷

of

definition

of

(E, s)

order

[B],

[B*],

is c o m p a r a b l e

the

edge

set

any

e , f • E,

the

interval

(e,f)

so

(e,f)

=

{[B] I [A] < [B] < [C],

:

{g I[B] I [A] <

But for any

of I.§9

To v e r i f y the

B •

interval

X

E

E is is c l o s e d

Say

only

< EIEB]

then

so in this

cas

~.) [A]

since

is c o m p a r a b l e

B

is proper.

to e x a c t l y

to give

a

is finite,

~I[B]

are

is i n f i n i t e

one of

G-forest

is a tree we m u s t

[e,f]

is finite.

~ there

{g • G I [A]

applies

that

S,

A , B e E,

Thus

E.

Z,

is b i j e c t i v e ,

for any

one of e

P(G),

~.

the p a r t i a l

e,f e E,

construction

subset

or e q u i v a l e n t l y

(6) that

to e x a c t l y

for any

and H,

not n e e d

It f o l l o w s

As

and by the

that

map

I A c Z}.

f,f*. X

s h o w that

with for

or e q u i v a l e n t l y

e : [A],

f = [C],

A , C • E,

B • E}

< [C], finitely

B •

~, g e G}.

many

such

g

since

< [C]}

(7) =

{ g e G I g[A]

< [B]}

n {g • G I g [ C * ]

< [B*]}

~ B n B*

71

DECOMPOSITION

by

2.6

(i).

finite The

Since X

stabilizer

of e a c h e d g e

is finite

G\X

is finite. Thus

the

so for any X

is f i n i t e ,

and h e n c e

G\E

of

~

~

subgroup

K

of

G[e]

=

e = [A],

then

G[e]

vertex

of

X

if and o n l y

K-subset.

In p a r t i c u l a r ,

Extend

to a (finite) By the

connecting over

H,

family which

In o r d e r result. each 3.3

can be

to a p p l y

LEMMA.

Hv

=

for each

1.8.8

is

2.6

connected

subgroups

says

that

or

graph

fixes

right

a vertex

K-subset.

< e},

Thus

so if

K

fixes

a

is an a l m o s t

right

every

A• ~

is an a l m o s t

right

H

fixes

connected

a vertex

of

transversal

theorem,

1.6.1,

a (finitary)

seen to have

inductively

X, in

say X

choosing

we n e e d

(Gv I v • V)

desired

G

a of

G

properties.

a rather of

v H.

for the

decomposition

all the

and

are b o u n d e d ,

K

ge*

(i).

(it),

A • ~

gives

3.2

so the

by

is an a l m o s t

by 2.6

Suppose

v • V,

G

is

finitely

be a f i n i t a r y Gv

almost-right-invariant

are

finitely

m a n y ucosets

right gG v

generated

decomposition

is f i n i t e l y

for each only

[e,f]

D

combinatorial

we w r i t e ,

for

H n G v.

(Gv,qe I v e V, e E E) Then

is f i n i t e

stabilizer

For any d e c o m p o s i t i o n

v • V,

that

if e v e r y

structure then

X

G[e]

~ A

or e q u i v a l e n t l y ,

G-action.

of

{g • G I ge < e

K-subset,

vH

G,

if e v e r y

But by 1.9.1, say,

we d e d u c e

is finite,

of the edge

if and only

§3

is a tree.

because

orders

THEOREMS

generated H-subset

that

lie

over of

H G

over A

and

let

over

H.

Hv, of

in n e i t h e r

and

G, A

there nor

A*.

72 III

DECOMPOSITIONS

Proof.

Take a set

assume

without

and that write By

S

that

loss of generality

for each

e e E,

Sv = S n G v ,

and

11.3.3,

Suppose

G v = G'v,

gG v

P(G,Sv), joining

A

S

subgraph

that

finitely

S-H

that

lies

(gGv)Sv

generated

generated

in n e i t h e r

so meets

6A.

and the cosets

gG v

are disjoint,

But

over A

is connected

A*,

H,

and

u u Gv, V For each v e V

Ge, G~e.

so is finitely

over

~ {qeleeE}

for the subgroup

to

of them.

G

contains

G'v

is a coset

the

generates

by

Hv.

nor

A*.

so contains

~A

is finite

so there

S v.

In a path

by 2.3,

can be at most

16AI

D

We are now in a p o s i t i o n

to prove

the main

decomposition

theorem. 3.4

THEOREM

over right

H.

(Dunwoody

Let

~

H-subsets

(Gv I v • V)

(8)

Let

finite

set of

infinite

of

G.

H

S

S-cuts.

connected

are p r o p e r

further

connected

S-cuts

that

(and right

H.

v E V.

Then

the elements

of

~

is a by their

that the elements

H-subsets),

is a finite

~

full

one vertex

of

G

over

decomposition.

H,

of

and by

set of proper

H-subsets).

decomposition

the trivial

over

(and right ~

decompositio n

for every

we may assume

S-cuts

assume

generated

A • ~,

Gv-SUbset

finitely

components,

Take any finitary for example

for eaoh

right

G

is finitely

has a finitary

By replacing

connected

2.6 we may

G

such that

generate

G

set of a l m o s t - r i g h t - i n v a r i a n t

Then

is an almost

Proof.

Suppose

be a finite

over

A

[79]).

(Gv I v e V), If each

73

DECOMPOSITION Ae¢

satisfies

not

satisfying

(8) t h e n we are (8) c h o o s e

Now consider left

cosets

any

of

G

glGv,...,gnGv .

in

v

g E G

(9)

g I A n gg-~A,

has

all

its

so one of the

for any

one of the

of

A,

has

situation,

3.2 g i v e s

w E Wv),

*v

w e W v.

right

of the such

form

B

finitely nor

i,j

many

A*,

say

= 1,...,n,

g-iA* n gg-~A*

xB,

B ~ ¢,

is an a l m o s t right

B ~ A.

right

Gv-set ,

intersection

i=1 ..... n },

Gv-set,

and

with

G v.

and any

g ~ G v,

each

applies

w c v W ), V v For e a c h v E V, each

where B E ¢

An giGv ,

w ~ W.

of

Hence

v c V,

decomposition the that

gw

are

In this

Gv

over

right

H v,

Gw-SUbset

i = 1,...,n,

is an

so is

n i~l(A n giGv )

to each

G v.

is an a l m o s t

each

for e v e r y

in

decomposition

B e Cv

is,

) u

B *v n gC *v

the c o m p l e m e n t

a finitary

That

Gw-set

B*V n gC,

1.2 to get a f i n i t a r y

(gwGwg~I

A

g-IA* n gg~A,

or c o f i n i t e

denotes

such that

The p r e c e d i n g by

only

sets

= {g~A n Gvl

n ( A - (i~lgiGv)

Gv

are

in n e i t h e r

is an a l m o s t

B n gC*v,

where

almost

A~¢

the

four sets

is finite,

for e v e r y

(9)

finite

B,C E Cv

B n gC,

(Gwl

each

among

under

so for each

four

components

four sets

in p a r t i c u l a r , Thus

of the

there

lie

g'iAn gg'~A*,

infinite

By the m i n i m a l i t y

3.3,

that

if not,

is m i n i m a l

is full,

one

v'

By

G

Now

and each

finished;

one that

v E V.

§3

THEOREMS

=

A.

so we m a y of

G

over

suitable

is an a l m o s t

expand

all the

H,

elements

right

G -set v

of

G.

is an

74 III

DECOMPOSITIONS

almost

right

Gw-SUbset

for every

almost

right

Gw-SUbset

for every

right

Gw-SUbset

is also an almost

So with the new finitary increased

the subset

is finite, a finitary

4.

of elements

decomposition

G

of

G

H

A

is an

right

of

¢

G

g w G w g ~ -subset.

over

satisfying

H,

we have

(8).

Since

in this way eventually

over

H

of

with the desired

gives property,

0

WITH

be a group,

of

to expand

is proved.

THE RELATIONSHIP

Let

decomposition

continuing

and the t h e o r e m

moreover,

w e V W v. Since the elements V any element of ~ that is an

are a l m o s t - r i g h t - i n v a r i a n t , almost

w E Wv;

DERIVATIONS

a subgroup

of

G,

and

R

a nonzero

ring. Given any right a decomposition decomposition group

Gv

(G v I v E V)

admits

is finitely

P,

generated

decomposition

DEFINITION.

way,

the set

module subring

G,

we shall

d:G ÷ M,

and

say that the

of

over the ring EndR[G]M.

if

is that,

d:G + P

which

d

to each vertex

M

of

amount

then

G

G

has a

of notation.

R[G]-module.

of all derivations

Thus,

d,

such that

d.

be a right

E = End~[G]M ,

for any projective

is a derivation

admits

a substantial

Let D(G,M)

chapter

over the kernel

We begin by fixing 4.1

of

and a derivation

if the r e s t r i c t i o n

of this

R[G]-module

finitary

d

M,

is an inner derivation.

The m a i n result right

G-module

d:G ÷ M

In a natural is a right

and in p a r t i c u l a r

for example,

D(G,R[G])

over the

is a left

75 THE R E L A T I O N S H I P

WITH

§4

DERIVATIONS

R[G]-module. The map as image

ad:M ÷ D(G,M),

the

E-submodule

inner derivations, M

fixed by

write

~G

G. for

Z

H-derivation.

intermediate

g~G 0

if

G

is infinite,

G ÷ M

H

D~(G,M)

DH(G,M)

of

that

R[[G]],

eD(R[[G]]),

be inner. (10)

is,

of all

of all elements R[G]~ G,

of

where we

R[G]o G,

where

is called an G + M

denote

it by G ÷ M

a left

the map

for all way,

a e R[[G]] we have

and we shall denote is then a left

ad:D(R[[G]])

÷

Notice

R-bimodule

s e S,

R[S]

RIG]

will be called ~(l-g)

of

ad ~,

R[G]-linear

D(G,R[G]).

the

g R[[S]]. as

RIG]

almost-riGht-

c RIG].

R[[G]].

g ~ e(l-g), it by

RS;

will be w r i t t e n

containing

R[G]-submodule

G ÷ R[G],

The

whose

D~(G,M).

for the

R[G]-bimodule

g ~ G,

DH(G,M).

D

R[[S]] rs e R

is an

the set of all a l m o s t - r i g h t - i n v a r i a n t

clearly

There

H

will be denoted

In a natural as

on

of all derivations

An element

denote

consisting

=

H-derivations

and we shall

we write

if for all

derivation,

vanishes

+ InD(G,M).

Z rs.S = ~ s.r s. S S We view R[[G]]

~(R[[G]])

which

is inner

S

(rs)s,

and has

R[G]/R[G]o G.

D(G,M),

For any set

R[G] G

is finite,

to

invariant

=

G

restriction

sub-bimodule.

InD(G,R[G])

if

E-submodule

:

E-linear,

D(G,M) MG

Since

The set of all of

of

g

=

A derivation

is right

the set

ad I:G + R[G].

R[G]~ G

element

InD(G,M)

For example,

~G =

E-submodule

m,

and has kernel

I

we see

m~ad

Let elements For any

is clearly although map

a

it need not

76 III

DECOMPOSITIONS

The kernel R[[G]], subset

that

is,

elements

of

G,

we write

A

R[[G]].

Then the kernel

see that x:R[G]

4.2

of this map consists

(10)

÷ R

of all "constant"

of the form

E r.g, geG for the element

XA

of (10) its

For any

be the coefficient-of-x

The left

ad:D(R[[G]])

+

map

r e R.

of For any

Z g of geA = RXG. We shall now

R[G]XG

is in fact surjective.

PROPOSITION.

elements

x e G,

let

~ r .g ~ r x. geG g

R[G]-linear

map

D(G,R[G]) ^

has an

R-linear

left inverse

a__nn R[G]-module Proof.

isomorphism

For any

~ [xd]x.x, xeG D(R[[G]])/RXG =

d ~ D(G,R[G]),

( Z [xd]x.x)(1 - g) x~G =

d ~

=

Z xEG

2 [(xg)d]~g.xg xcG

and any [xd]x.x

-

-

g

and so induces D(G,R[G]).

G,

e

~ xeG

[xd]x.xg

[xd.g]~g.xg

E

xeG

^

=

~ [gd]xg.xg xeG

=

gd.

The preimage RXG + RIG]x1, write of 4.3

in

DH(G,R[G]) REMARKS.

by any

s e G

D(R[[G]])

and and

of

is a derivation

of finite

D~(G,R[G]),

or eofinite

DH(R[[G]])

=

R[G]6 G support.

for the preimages

in

is We ~(R[[G]])

respectively.

of the elements

pairwise

changes

InD(G,R[G])

~(R[[G]])

We think

Z rXAr, the Am reR formal sum lies in

d

17

the elements

~H(R[[G]])

since

disjoint

of

R[[G]]

and having

as formal

union

G.

sums

Such a

if and only if right m u l t i p l i c a t i o n

only finitely

many coefficients,

and right

77 THE R E L A T I O N S H I P

multiplication that ArS

is,

by any

Arh = A r

@ Ar

for all

any generating

(11)

Thus

finitely

S

many of the

~H(R[[G]]) form

r e R, h E H, ArS

= Ar

G

over

H,

then

Ar

for almost in

If

so is

sum.

G

is s e n e r a t e d

Proof.

The first

the fact that

part follows

which means

in which

senerated

R-module

about

THEOREM

generated there

exists

ever~

element

Proof.

taking

H.

only

case our formal

matter

over

H

H-subset

then of the

of

G.

and the second part + R[G]6 G.

to t r a n s l a t e

of

[79]).

Suppose

that

Then for any finite

a finitary

For each

derivation,

For

from

D 3.4 into a

derivations.

(Dunwoody

over

that

by the elements

from 4.3,

DH(G,R[G])

It is now a s t r a i g h t f o r w a r d statement

r e R.

0

as left

=

all

I gEAr-ArS'1})-

(11),

is finitely

D~(G,R[G])

s E G,

F(G,S),

XA, A an a l m o s t - r i g h t - i n v a r i a n t right + DH(R[[G]]) = DH(R[[G]]) + RIG]x1.

Also,

4.5

and for every

are nonempty,

(finite)

PROPOSITION.

none of the coefficients;

U ( u {(g,gs) sES-H rER

finite

§4

DERIVATIONS

and

of

=

is

sum is a genuine

4.4

r E R,

set

S - H

changes

for all

U 6A r rER

if

h E H

WITH

decomposition

of

G

subset G

is finitely ~

over

of H

D~(G,R[G]), that

admits

~.

d E ~

we may subtract

and so assume

~ ~ DH(G,R[G]).

that

d

a suitable

vanishes

By 4.2 and 4.4,

on

there

H.

inner Thus we are

is a finite

set

78 llI

OECOMPOSITIONS

of a l m o s t - r i g h t - i n v a r i a n t

right

each

d ~ ~,

ad( Z rA,dX A) Ae} is a f i n i t a r y d e c o m p o s i t i o n

there that

each

every

d

H-subsets

=

element

v e V,

A

of

~

from which

D~(G,R[G])

for e v e r y

G,

for c e r t a i n (Gv I v e

is an a l m o s t

it is e a s y

v e V,

of

such that

r A , d ~ R.

V)

of

right

G

By

over

ad X A

and h e n c e

the

same

following

formulation.

such

for

lies

is true

3.4,

H

Gv-SUbset

to see t h a t

for

in

of e a c h

V

element

of

~.

For C h a p t e r

4.6

THEOREM.

d
l_~f

P

of

d, t h e n

admitting

Proof.

d,

There

we may a s s u m e

the

is a p r o j e c t i v e

is a d e r i v a t i o n

the k e r n e l G

IV we n e e d

such

there

that

is no h a r m that

P

G

exists

and h a v i n g

ker

R[G]-module,

is f i n i t e l y

a finitary

d

an

generated

vertex

R[G]-summand

R[G]-module.

to

generated

over

ker

d,

the

image

finitely

generated

submodule

of

P,

so we may a s s u m e

finite

over

rank.

ker d

Now c o n s i d e r exists G v.

Now

admitting

we m a y a s s u m e

REMARKS.

p

that

a finitary

component

group

of

p = 0,

t h e n by a d j o i n i n g

4.7

p

In p a r t i c u l a r , then

each

the v e r t e x

an e l e m e n t

infinite

4.5 g i v e s

P

Gv

d,

ad p ker

Gv : ker d

a new edge ker d

by

d. If

and a n e w v e r t e x

appears

Let us e x a m i n e

what

d

G

lies

says

is

in a that

d.

P of

agrees If

with

ker

ker d

G d.

There

d

d

on

is

is f i n i t e ,

to the

decomposition,

as one of the v e r t e x

4.5

so

so a d m i t t i n g ker

of

$roups.

decomposition

containing

such that

is f i x e d so

of

over

P,

Since

finitely

has

of

and

decomposition

as o n e of the

in a d d i n g

is a free

rish t

in t e r m s

groups.

of g r o u p s

D

79 THE RELATIONSHIP

acting

on trees.

decomposition tree,

Thus,

of

G,

cf 1.4.4.

into a right

g • G.

We shall

so the edges

I v • V(Y),

e e E(Y)

be the c o r r e s p o n d i n g

has a left action

action by setting

x.g

on

X

= gl.x

)

be any

standard

which we for all

x • X,

view

=

of

G

X

§4

DERIVATIONS

(Gv' qe

and let

Then

convert

E(X)

let

WITH

V e•E(Y)

X

Ge\G ,

V(X)

=

V v•V(Y)

GvXG ,

have the form

G leg

-I GTeqeg.

G g

A

For each

y e Y,

we write

that

Gyl.

Then

is,

X(v0,~0_):G original

module

factors

decomposition

4.8

there

d.

is a

cf 1.§1,

derivation,

-derivation

through

representative

÷

R,

1.§5.

Our

to a right

see

R[G]

if and only if our original

D

the a u s m e n t a t i o n

and is called

E

u~U

map,

the a u ~ m e n t a t i q n

r

G-set

.u

~

u

U,

there

is a right

Z

ueU ru'

and the kernel module

X,

Gv0-derivation

and we shall

d:G ÷ M

X(v0,v0_)

in

map

cu:R[U]

called

v0

this

For any right

DEFINITION.

R[G]-linear

admits G

admits

v0 • V(Y)

g ~ X(v0,v0g),

decomposition

M,

for the natural

for any

÷ R[E(X)],

later that any other

y

of

U

is denoted over

R.

mR(U), Q

80

III

DECOMPOSITIONS

4.9

PROPOSITION.

G-set,

and

z

Let

M

be a r i g h t

an e l e m e n t

dz:G + m R ( U ) ,

g ~

of

z - zg,

image

of thi..s map

In p a r t i c u l a r , map

InD(G,M)

there

The

induces

HOmR[ G] (mR(U) ,M)

and the

U.

+

is

EndR[G]M-linear

e

n

Since

we see t h a t

dz

for any

i n n e r on e a c h that

n

chosen

agrees

so that

m

ad(z-

+ M,

G u. with

= 0 and

on

on

for e a c h

vanishes

u e U Gu,

for e a c h

Gu

d:G + M

For e a c h ad m u

u

u)

let

EndR[G] M-linear

D~(G,M).

dz-e

Conversely,

on e a c h

d

with

e:WR(U)

G u.

is i n n e r

such t h a t

agrees

dz.~,

right

ucU Proof.

~+

map

n D~(G,M). uEU u

iS a s u r j e c t i v e

÷

a right

Z

a right

DG(G,M) z

U

G -derivation

D(G,M) ,

HomR[G](mR(U),M)

@

R[G]-moduie,

on

be a

there

and

and is

Gz-derivation

exists

and the

u e U,

Gz

u E U,

mu

m uE M

m a y be

g e G,

Z

mug

=

in

mug

U

+ gd.

for the

that

d

G-action,

agrees

msg

=

map

~:mR(U)

msg

(To see this,

with

+ gd

and

ad m s

for e a c h

÷ M,

is in the

=

(z

-

image

(We r e m a r k

zg)a

=

t e r m of a l o n g e x a c t

Gs,

g E G.)

to a t r a n s v e r s a l

s E S,

choose

and

m z = 0,

Thus,

there

ms e M

and t h e n is an

~

-m z + m z g

=

gd

as d e s i r e d .

foregoing

Ext

z

( E m .r ). For any uEU u u = E m r if E r u u~U ug u ueU u Further, for any g e G,

of our map,

t h a t the

for e a c h

on

( E u . r u) ueU (mug - g d ) r

E mugr u = E uEU ueU is r i g h t R[G]-linear. (g)dz'e

extend

amounts

sequence.)

since

mz

=

S so

define

R-linear g e G, =

0,

O,

so

so

d

D

to the e x a c t n e s s

at one

81

THE RELATIONSHIP 4.10

COROLLARY.

For any right

HomR[G](mR(HXG),M)

4.11 =

REMARK.

= DH(G,M)

R[H\G]

=

R@R[H]R[G].

Thus,

HomR[G](R[H\G],M)

=

modules.

4.12

R[G]-module

as right

HomR[H](R,M)

M,

EndR[G]M-modules.

=

M N,

as right

EndR[G]M

D

PROPOSITION.

Let

for the qprrespondin$

v0 e V(Y).

M

be a right

R[G]-module,

)

be any decomposition

standard right

G-tree~

Then there is a s u r j e c t i y e right

(12)

HomR[G](R[E(X)],M)

÷

DG(G,M) v0

~

X(v0,v0

^

In particular, @

~ eEE(Y)

Proof.

From 1.1.1,

e ~ ~e-Te,

M Ge

÷

- ~0g,

In light of 4.12,

and choose any

n vcV(Y)

map

D~(G,M) ~v

)'~.

right

EndR[G]M-linear

map

R[E(X)]

= ~R(V(X)),

In particular,

so the result

follows

from 4.11.

from 4.9.

D

4.5 may be viewed as saying that there exists

G-tree

finite,

such that each element

with finite edge stabilizers,

from an

It is interesting

Write

n D~(G,M). v E V ( Y ) ~v

a right

derivations,

X

G.

^

R[G]-modules.

The second part follows

of

and

EndR[G]M-linear

we have an isomorphism

of right ~+v0

n

there is a surjective

InD(G,M)

X(v0,~0g)

§4

@ R[H\Hg] = @ R~R[H]R[Hg] HgeH\G H\G for any right R[G]-module M,

(G v, qe I v E V(Y), e e E(Y) X

WITH DERIVATIONS

of

R[G]-linear to examine

~ map

arises,

and with

X/G

modulo inner

R[E(X)] + RIG].

these derivations

in greater detail.

82

III

DECOMPOSITIONS

Thus, and

let

X

suppose

linear

map

be Ge

a right is

which

G-tree,

finite.

sends

e

and

Let to

let

e

~:R[E(X)]

aG

and

be

any

÷ RIG]

vanishes

edge

of

be the

on all

X,

RIG]

other

e orbits

in

E(X).

derivation of

d

in

F o r any

d:G + R[[G]].

+1 [xd]x

RIG],

=

I -1 0

vertex g ~

For

of

X(v,vg)~.

each

X,

we h a v e

a

Now

consider

the

preimage

x E G,

if

v ÷ ex ÷ vx,

or e q u i v a l e n t l y ,

vx -I ÷ e ÷

v

if

vx ÷ ex ÷, v,

or e q u i v a l e n t l y ,

v ÷ e ÷ vx -I

otherwise.

To c o n f o r m

with

G[e,v]

=

{ x e G I e ÷ vx-1}.

v + e,

separately,

5.

v

the

notation

we

used

in

I.§8,

we w r i t e

By c o n s i d e r i n g

see

from

4.2

that

the d

=

cases

e ÷ v,

- ad X G [ e , v ].

ACCESSIBILITY

Let

G

be

a group,

H

a subgroup

is

loosely

of

G,

and

R

a nonzero

ring. This

section,

considers in this

which

various

section

A derivation

will is

A decomposition one

of the An

both

consequences

vertex

be

used

These

and

A*

concepts

4.5.

in C h a p t e r

called

outer

of

is c a l l e d

G

if it

on D u n w o o d y

[79],

None

results

o f the

proved

IV.

is not

proper

inner.

if

G

does

not

is

called

occur

as

groups.

almost-right-invariant A

of

based

subset

are

infinite.

are

related

as

A

of

follows.

G

proper

if

83

ACCESSIBILITY

5.1

THEOREM.

The f o l l o w i n $

(a)

There exists stabilizers, fixes

(a') There exists

four statements

an action of such that

some v e r t e x of

§5

G G

are equivalent.

on a tree

X

with

fixes no vertex

of

finite edge X,

and

H

X.

a proper decomposition

of

G

over

H

with

finite edge groups. (b)

There exists stabilizers, fixes

an a c t i o n

of

such that

some v e r t e x

v0

G G

X/G

has p r e c i s e l y

(14)

There is an "infinite is a t r a n s v e r s a l

that

H c G

(b') At least one of the (15)

G

=

(16)

G

=

(17)

G

is infinite

(c)

for the

these

locally

and

H holds:

G-action~

_in _

X

and such

holds. H ~ A,

A > C < B.

C finite, finite,

i m p l y the f o l l o w i n g

and

H ~ A. H

is finite.

two statements

which

are

to each other.

(e') There exists

a ~roper

H-subset

of

and its converse,

G ÷ RIG].

almost-ri~ht-invgriant

senerated

(and here

(a)~(a'),

H-derivation

right

G.

is finitely

are e q u i v a l e n t

(c)~(c')

v0,e ~1 I ,Vl,...

path"

A~B, C finite, C H N N < e , B : C ÷ A;t>,

an o u t e r

Proof.

X,

and one of the following

following

There exists

I_~f G

X,

finite edge

c G c ... is a strictly a s c e n d i n g el e2 (of finite sroups, whose union is all of G).

chain

equivalent

with

one edge.

whieh

Further,

X

fixes no vertex of

o_~f

(13)

on a tree

(14),(17)

(b)~(b') 1.6.1,

by the remarks

over

follow

H

then all six statements

do not occur).

from the s t r u c t u r e

theorem

1.5.3. made

in 4.3;

for if

d:G + RIG]

is

84

Ill

OECOMPOSITIONS

an outer derivation pairwise

ad( ~ rx A ), where the A are reR r r almost-right-invariant right H-subsets of G,

disjoint

then

such that the union of these is again

if

H-subset

G

of

(a)~(b). that

is all of

to choose A

having

then

=

of

a natural

G-action

E

two cases:

edge

X,

e

of

the former

with

1.8.4

subgroups

there

of

ell ,ei2 ,...

and take

to be the

is as in (14). (b) ~ ( c ' ) .

shows

since there

of

right

H-derivation.

exists

an "infinite

= Ge2 ~ ...

the graph

(This

is an

of 1.§9.)

of

G

We

shifts X.

an In

is as in (13) g);

in the latter

path"

an ascending

union

of

observe

an edge of X

is fixed by

G-subset

g

shifts

and then

such that

E.

construction

G

is all of

~1 ,Vl,--v0,e I chain

G.

of

We choose

a

H c G eil c Ge.m2 c ... E(X)

it generates.

Then

(b) holds. then

right

n~lU(Ge2 n- Ge2n_l)

H-subset

does not stabilize

is an edge

is proper.

is a tree with

on

X

If (14) holds

G

X

some element

whose

Thus

almost-right-invariant then,

of

G

subsequenee E

the

It

to the set of connected

G-action

E = eG,

H ~ Gv0 = Gel ~ Gvl

(finite)

the

or no element

shows

equal

either

case we take

union

we can construct

It can be seen that

(since by 1.8.3 no vertex case,

whose

is an outer

E(X)

to illustrate

now consider

right H-subset.

almost-right-invariant

V(X)

extending

of any subfamily

To see the converse,

of

and

X - ~.

example

a subfamily

( b ) ~ (a).

E

components

interesting

and the union

ad XA:G ÷ R[G]

G-subset

E(X)

G

is a proper

Clearly

for any

=

an a l m o s t - r i g h t - i n v a r i a n t

is an easy m a t t e r Conversely,

d

e

shifted

of

G.

any vertex

is a p r o p e r

If (13) holds of

by an element

X, of

1.8.4 G,

and then

85

ACCESSIBILITY G[e]

is an a l m o s t - r i g h t - i n v a r i a n t

by 1.8.6,

and is proper

Finally,

if

by 4.5. more

G

(Alternatively,

in the spirit

H = 1.)

G

that

G

is finitely

is (finitarily)

(equivalent) finitary

decomposition

"finitarily"

over

over

H.

anywhere

is i n d e c o m p o s a b ! e

usually

that

of

G

(c') holds.

H

then

3.2 and 2.8,

(e) = (a')

which

of ( c ' ) = ( b ' )

is

for

H;

(c) fails

=

H

R[H]

as a subset h ~ l-h,

of

that

We shall

if the above is,

G

H" if

DH(G,R[G])

if

G

say

six

has a proper

is (finitarily)

is finite

or 4.2.

We shall

then

H = 1.

+ R[G]6G,

D~(G,R[G])

with no o t h e r

G

:

we see that R[G]~ G,

so the

for indeeomposability. more

general

of

D(H,R[G]).

(In particular,

an element

of

RIG],

version

is a subring

is thus

H.

H

otherwise

by 11.3.5

"over

We shall need a slightly Since

over

For example,

if and only if

is a criterion

over

since we shall be dealing

over

D~(G,R[G])

generated

(We shall not be using the q u a l i f i e r

omit the q u a l i f i e r

Since

latter

H-subset

over

[68] proof

are satisfied,

type of decomposability.)

5.1

generated

decomposable

conditions

indecomposable

G

It follows

one can use

Stallings

right

[]

Suppose that

of

almost

by 1.8.7.

is finitely

§5

of this.

we may view

D(H,R[G]).)

D(H,R[H])

6H:H ÷ R[H],

86

Iii

DECOMPOSITIONS

5.2

LEMMA.

embedding

Let

K

D~(K,R[K])

be a subgroup

of

G

containing

~ D~(K,R[G])

induces

H.

The

an injective

RIG]

linear map which will be treated as an embedding

D~(K,R[G]).

+ R[G]@R[K]DH(K,R[K]) If

K

Proof.

is finitely

generated

R[G]@R[K]D~(K,R[K])

=

gKeG/K

is inner on

H

fact it suffices generates

K

then equality holds.

@ gR[K]@R[K]D~(K,R[K]) gKeG/K

if and only if each derivation

d:K ÷ R[K]

has image lying in a finite sum of

gR[K],

to consider

finitely

d:K + RIG],

5.3

H

gKeG/K

Equality holds

proof.

over

we have

over (K)d ~

only

H-derivations

H.

Then for any

~ (s)d.R[G], seS

d.

that and in

Suppose

S

H-derivation

which completes

the

0

INDECOMPOSABILiTY

containing

H

Let

CRITERIA.

and finitely generated

K over

be a subgroup H.

K

is indecomposable

(b)

DH(K,R[K])

=

R[K]6 K (=

R[K]/R[K]oK).

(c)

DH(K,R[G])

=

RIG]6 K (=

R[G]/R[G]OK).

(d)

DH(K,R[G])

=

R[G]OH6 K

Proof. (b)~=~(c)

(a)~=~(b)

over

H.

(=

by the remark

R[G]OH/R[G]o K) . preceding

5.2.

since by 5.2,

n~(K,R[GI)/R[G]~ K = ~[G]~REK](n~(K,REK]/REK]~K). (c) ~ ( d )

since

DH(K,R[~]) n RIG]6 K

(d) ~ (c)

since

D~(K,R[G])

=

=

R[G]OH6 K.

DH(K,R[G])

G

Then the following

are equivalent. (a)

of

+ R[G]6 K.

D

87

ACCESSIBILITY

Let us r e c o r d one c o n s e q u e n c e

5.4

COROLLARY.

If

and is f i n i t e l y

=

In p a r t i c u l a r ~ then

Suppose

=

that

over

H,

G

D(G,R[G]).

is finitely

such that,

Hv

is finite

H,

over

Hv;

(Here, for

H

if

that

G

is,

If

No example

H"

if

is k n o w n where

and known to be i n a c c e s s i b l e that every

finitely

p r o b l e m remains

generated

over

H,

Then

v e V,

G

is

decomposition

Gv

Is

a sort of "atomic"

G

are finite,

is not a c c e s s i b l e over

H.

As usual,

each

over

H

we omit

H = 1. G

is f i n i t e l y

over

H.

group

Wall

generated [71]

is a c c e s s i b l e ,

over

H

conjectured but so far the

open.

We now turn our a t t e n t i o n relate

H.

has a f i n i t a r y

since the edge groups

v ~ VH.)

"over

over

for each

then it is said to be i n a c c e s s i b l e the q u a l i f i e r

H,

g e n e r a t e d and indecomp.osable

generated

(Gvl v e V)

decomposition.

that contains

~

over

indecomposable

G

and i n d e ¢ o m p o s a b l e

K is finitely

said to be a c c e s s i b l e over

of

D~(G,R[ G]). if

D~(G,R[G])

of 5.3.

is a subgroup

generated

D~(G,R[G])

then

K

§5

accessibility

to p r o v i n g

to derivations.

Dunwoody's

results

which

88

III

DECOMPOSITIONS

5.5

LEMMA

(Bamford-Dunwoody

be a d e c o m p o s i t i o n each edge

e

in

indecomposable

of

G

[76]).

over

star(v0) ,

H,

Ge

Let

(Gv,qelveV(Y),eeE(Y))

and let

v 0 e V(Y).

is finitely

then the r e s t r i c t i o n

map

If for

generated

D~(G,R[G])

and

÷ D~(G v ,R[G]) v0 0

is surjective.

Proof.

Let

the maximal star(v0) ,

d :G + RIG] v0 v0 subtree of Y.

the r e s t r i c t i o n

say it equals

from

clearly

to

agree

v.

on

the derivation e 4 star(v 0) de;

if

G e.

e e E(Y)

with

de

Let

d e

agrees

on each

so

d

then

dv

d

We can now prove

5.6 over

THEOREM H.

D~(G,R[G])

H;

if H.

G

are both

vH ~ v 0

is finitely

Y,

(c).

If

of

T

so is

Thus

ad meq 2

d:G + RIG]

for agrees

we may take such that

qe

to

GVH

and

dv0

is inner on

H

Suppose over

G

d

me

is inner on

characterization

generated

and consider

- (g)d e.

the lemma.

in

dTe

dvH

This proves

is accessible

and

then

vH = v 0

[79]).

d%e

such that

each

(c),

define

inner and hence

e

in

is the geodesic

is inner by 5.3

and sends

for

lying

v ~ v0,

T,

(gqe)dTeq2

for the edges

our first

(Dunwoody

Then

g ~

T

T

is inner by 4.9

of

m e e RIG]

Gv,

If

is inner on

e

is a derivation

e E E(Y)).

is inner on

Ge

of

be any edge of

de

further,

e

e~1,...,e nCn

dTe

then

there

(v e V(Y),

e

and

m e = 0. By 1.5.2 with

where

we may choose Ge;

to

Write

For each vertex

de:G e ~ RIG], then

each

dv0

For each edge

e E star(v 0)

on

of

to be ad el

v0

For each edge

ad ~, ~ E R[G].

dv:G v ÷ R[G] T

be any derivation.

O

of accessibility.

is finitely

H

if and only

as left

R[G]-module.

generated if

89

ACCESSIBILITY Proof.

=

Suppose

(Gv,qelV~

V, e E E)

v0 e V

surjeetive

left

over

H,

say

v e V,

Gv

is indecomposable

of

G

over

over

H

H v.

and apply the last part of 4.12 to get a R[G]-linear map from

RIG]6 G @

=

is accessible

is a finitary decomposition

such that for each Choose any

G

§5

~ R[G] Ge eeE

RIG]6 G @

@ R[G]OGe eeE

onto n D~(G,R[G]) V v

This gives

=

n D~(G,R[G]) V v

=

D~(G,R[G]).

IEI + I

Suppose

elements

D~(G,R[G])

finite set

~.

(Gv I v E V)

of

by 5.4

that generate

is generated

D~(G,R[G]).

as left

R[G]-module by some

By 4.5 there is a finitary d e c o m p o s i t i o n G

over

H

such that for each

v E V

the

r e s t r i c t i o n map

(18)

D~(G,R[G])

÷

D~(Gv,R[G]) V

carries

~

to

the image of D~(Gv,R[G])

R[G]~Gv. ~

=

But by 5.5,

is a generating R[G]~Gv.

set,

(18) is surjective, and therefore

Now 5.3 shows

Gv

is indecomposable

--V

over

Hv,

which proves that

G

so

is accessible

over

H.

0

90 DECOMPOSITIONS

III 6.7

COROLLARY.

Then

G

Suppose

is accessible

finitely

generated

Proof.

Recall that

G

over

H

as left

DH(G,R[G]) n R[G]~ G

is finitely

D~(G,R[G])

only if

=

R[G]OH~ G. ~

is finitely

DH(G,R[G])

is.

With this result,

over

H.

DH(G,R[G])

i__{s

R[G]-module.

D~(G,R[G])/R[G]6 G 80

if and only if

D~(G,R[G]) :

generated

DH(G,R[G])

+ R[G]~ G

and

Thus DH(G,R[G])/R[G]~H6 G

generated

as left

R[G]-module

if and

D

the generality

ceases to be important,

of

R

being arbitrary

and we turn to the case

R = ~

and

study the abelian group D~(G) where

~

[79],

then by 5.4,

where

D~(G)

PROPOSITION.

indecomposable

If

over

H

DH(G)--f_ZZ

ZIG] -module with trivial in Bamford-Dunwoody

A(G,H).

H

is finitely

generated

=

D~(G).

G

By 5.3,

+

is finitely

G-action.

[76] in the

version was introduced

it was called

in

indecomposable

Generated

over

H

and

then i_~f G

IG I

Proof.

D~(G,~[G])

and the relative

We remark that if

5.8

~[G]

group was studied

H = 1,

Dunwoody

~

is viewed as right

This abelian case

=

if

DH(G, 77[G])

is infinite G

is finite.

=

77[G]~ G

--

77[G]/77[G]OG,

so

91

§5

ACCESSIBILITY

we have

a presentation

77[G]

.,

77[G]

,

then

gives

ZZ@ZZ[G] _

Applying

nG

,

~

>

D (G,77[G])

O.

a presentation D~(G)

~

0

wh e re O if

nG

G

Ii

=

G I if

Recall

that

0

÷

where

(E n g g ) e

ideal

of

G.

there

~

G

is finite.

is an e x a c t

~[G]

÷

=

is i n f i n i t e

Z ng,

Applying

~

sequence

~

and

D

+

~

0

is c a l l e d

_@~[G]D~(G,~[G])

the

augmentation

gives

a

presentation

where

d

t h e n we G

0

÷

~D~(G,~[G])

=

l@d.

can w r i t e

is f i n i t e l y

decomposition that

is,

there

that

for e a c h

Thus dO :

generated (G v I v ~ V) exist i,v,

x.

X0'v

of

l~V

d. n

each

if we are g i v e n a d O such that d0 -- 0 n Z w i d i, w i ~ ~, d i E DH(G,77[G]). If i=1 over H then by 4.5 t h e r e is a f i n i t a r y

1

G

over

c ZZ[G]

agrees

with

H

admitting

(i = 0 , . . . , n ; ad x.

l~V

on

d o ,... ,dn,

v e V) G .

finite

So for

V

= i=l Z ad w.x. on G v, that is, ad x0, v m z,v n - i=lZ w i x i , v e 77[G]~Gv. In p a r t i c u l a r , either

such

v,

or

(X0,v)e

= 0.

We r e c o r d

a special

case

Gv

of this.

is

92

III 5.9

DECOMPOSITIONS PROPOSITION

generated over m~ G = 0

in

(Dunwoody H.

[79]).

Suppose

G

If there is a positive

D~(G)

then

G

is finitely

integer

m

such that

has a finitary d e c o m p o s i t i o n

over

H

in which a!l the vertex groups are finite. Proof.

We are in the situation of the remarks preceding the

proposition,

with

for each

and since

v,

(Of course and

G

do

ad m.

=

Thus we may take

(m)e = m ~ 0,

each

Gv

x0, v

=

m

is finite.

it follows that in the above situation

H

D

is finite

is accessible.)

In order to use the abelian group accessibility

of

G

with the various

over

D. ( G ) M v

H

D~(G)

to measure the

we want some way of comparing

DH(G)

that arise from a decomposition.

V

Recall that for any subgroup

K

of

G

there is a natural map +

DH(G, 77[G3)

÷ DHnK(K, ZZ[G])

where the latter containment generated over

HnK.

is equality

In this case,

a canonical map

D~(G) + D HoK + (K).

5.10 PROPOSITION

(Dunwoody

generated oyer

H,

and let

finitary d e c o m p o s i t i o n if

Gv0 c G

sur~ective Proof.

[79]).

of

77[ G]@77[G]DHnK(K, Z~[ K])

if

K

is finitely

tensoring with

Suppose

G

G

over

then the canonical map

H.

~

induces

is finitely

(Gv,qe I v e V(Y), e e E(Y)) Then for any

D~(G) ÷ D~(G v ) v0 0

be a v 0 e V(Y), is --

and has nonzero kernel.

From 5.5 we know t h a t

surjective,

~

and tensoring with

is surjective.

D~(G,~[G]) ~

÷

D$(Gv_,~[G]) is v0 U then shows that DH(G) + D~(Gv) +

It remains to find an element in the kernel t h ~

0

g3

§5

ACCESSIBILITY

is not each

zero.

We may

assume

v ~ V(Y) - {v0} ,

is i n a c c e s s i b l e over

H

Gv

over

is e i t h e r

H v.

and a c c e s s i b l e

without

indeeomposable

over

H

Gv

that

for

Hv

or

over

is d e c o m p o s a b l e

t h e n we may e x p a n d

the

V

decomposition H

by 1.2 to get

but h a v i n g

group,

of g e n e r a l i t y

(For if some

V

over

loss

where

are both

g : 1

finite.

a new

finitary

some

conjugate

if

v 0 : VH,

Thus

Gg v0

decomposition

of

G

that

G

as a v e r t e x

v0

H n Gv0

and o t h e r w i s e

it can be seen

of

there

H n Gg v0

'

is a c o m m u t i n g

diagram +

DH(G,2Z[G])

'

DHOG(vGvo, 77[ G] ) 0

g

DHg(Gg,77[G])

',

DHgnGvg(Gvg0 ,ZZ [ G ] ) . 0

',

D nG~(G 0

tl

DH(G,~[G]) so the new Let cf

X

4.7.

decomposition be the

standard

For each

the right

edge

R[G]-linear

on all o t h e r

orbits A

can be used

in

an e l e m e n t

de

choose If

Y

and then so

in such

D~(G),

of

(q~1)d e

e.

Y,

let $

and

d

let

of the

to the

old.)

decomposition,

Se:R[E(X)]

sending

Then

:

de

and of

a w a y that

is not a tree,

d e ~ D~(G,~)

choice

of

is in the k e r n e l e

associated

of

E(X),

in place

to e

÷ RIG]

be

and v a n i s h i n g

°G e

:G ÷ RIG]

be the

^

X ( v 0 , v 0 _ ) e e.

de

e

mapping

derivation

Gv 0'

tree

,~[G]) 0

then

further,

on

since

GVH

~e

+ D~(G v ). v0 0 ~ 0.

there

is some

e

so we have

de

D~(G)

( X(v^ 0 ' ~e)

~ ~D~(G,~[G]),

is inner

vanishes

It remains

with

+ e^ + X(~e 'v0)qe ^ -I )~e and thus

de

~ 0

on to

qe

~ 1,

:

qG e ~

for this

94

III

DECOMPOSITIONS This

there

leaves

the case where

must be some vertex

in the geodesic e

such that

We shall

Y

v

of

e~l .... ,e~n G e ~ GTee

show that

for this

choice

v0

to

v

there

choose

v

e,

de ~ 0

of

restriction

de

to

Gv

maps

to

is

Gv

is

thus,

Thus,

is some t e r m

in

v = Te

E

.

D~(G).

A

ad X(v0,v) ,

ad eOGe;

~ G,

v0

so that

^

X(v0,v0_)

G

Gv0 ~ G v.

^

of

de

Since

such that

from

The r e s t r i c t i o n

D~(G) + D~(Gv) ,

Y

and we may

^

of

is a tree.

so the

under

to

V

(19)

eaGe6Gv

elGel~ G

in

D~(Gv).

V

Suppose G

this

is zero;

is accessible

we get a c o n t r a d i c t i o n

over

H

V

over

H

by 5.8,

~Gv

has

order

of decomposition,

so by 5.9 again,

V'

IGvl

in

D~(Gv).

By 5.9,

as follows.

so by our choice

V'

is i n d e c o m p o s a b l e

V

Gv But

is finite, G e c Gv,

G and

so

V

IGel < IGvl , ~ O,

so

IGel-6Gv ~ O.

as claimed.

Hence

(19)

is nonzero,

and thus

D

e

5.11 DEFINITION. let us define (rank(A/tA),

For any finitely

the size of ItAI),

Two such pairs

tA

will be compared

(al,a 2) > (bl,b 2) defines

where

A,

if

a well-ordering

aI > b I

generated

size(A),

abelian

to be the ordered

is the torsion

subgroup

lexicographically, or if

group

A, pair

of

that

is,

a I = bl, a 2 > b 2.

of the set of pairs

A.

(This

of nonnegative

integers.) Suppose group

G

D~(G)

decomposition

is finitely is finitely D

=

generated

over

generated.

(Gv I v e V)

of

H

and that the abelian

For any G

over

finitary H

we can then

v

95

ACCESSIBILITY

define

the d e f e c t

of

D,

defect(D),

§5

to be the

finite

descending

sequence •

s i z e ( n H ( G v )), vI 1 where such

s i z e ( D ~ ( G v )), v2 2

Vl,V2,...,v n that

Gv

....

is an e n u m e r a t i o n

is d e c o m p o s a b l e

over

of t h o s e

H v,

+

s l z e ( D H ( G v )) vn n elements

v

of

and the e n u m e r a t i o n

V

is

such t h a t

s i z e ( D ~ ( G v )) vI 1 (For e x a m p l e , proper

if

a

G

decomposition •

one term,

s i z e ( D ~ ( G v )) v2 2

~

is d e c o m p o s a b l e then

n = 1,

...

over

and

~

H

s i z e ( D ~ ( G v )). vn n but

D

defect(D)

is not

consists

a

of

+

slze(DH(G)).)

Two such

sequences

(pl,...,pn)

>

be c o m p a r e d

(ql,...,qm)

there Pi+l

will

exists

lexicographically,

that

is,

if

i < n

such t h a t

Pl = q l ' ' ' ' ' P i

: qi'

> qi+l'

or

n> m (Again,

this

Suppose

Pl

= ql'''''Pm

say that

Let

us see h o w the d e f e c t s

say that

corresponding Hv

as in 1.2

to get a new f i n i t a r y

= ( g w G w g ~ l w e V W v) of G over H. H e r e we V D' is an e x p a n s i o n of D, a l t h o u g h to be p r e c i s e

should

over

D

D'

one

If the

= qm"

is a w e l l - o r d e r i n g . )

n o w t h a t we e x p a n d

decomposition shall

and

is not

D'

can be o b t a i n e d of

D

decomposition

proper

and

by an e x p a n s i o n

D' Dv

compare. =

t h e n the c o n t r i b u t i o n

Let

(Gw I w ~ W v) of

v

of

to

D. v e V.

of

Gv

defect(D)

96

III

DECOMPOSITIONS

is the

same

namely,

as the t o t a l

size(D~(Gv))

contribution

if

Gv

of the

we W v

to d e f e c t ( D ' ) ,

is decomposable over

Hv,

and

V

nothing

if

Gv

decomposition

term,

is i n d e e o m p o s a b l e of

G

over

v

size(D~(Gv)) ,

to

H

over

then

v

H v. v

defect(D),

If

Dv

is a p r o p e r

contributes

exactly

one

w h i l e any t e r m s c o n t r i b u t e d

V

to d e f e e t ( D ' ) the n a t u r a l

(20)

by the

D (G v) + D (G w) V W

see the

and has

isomorphism and we h a v e

are b o t h

finite,

in (20),

equality,

D'

decomposition

Dv

of

D.

improper

expansion

of

D

H

t h e n the

followin$

D~(G)

(b)

G

(c)

(Terminal

of

D

over

5.10,

If

either

so 1

Hw

in

expansion

D

H n gwGw~

if for some D'

that

if

is an D'

= defeet(D'),

defect(D)

G

and

(20).)

of

shown

(To

w = wH

and

and o t h e r w i s e

then

G

Every

is,

no e l e m e n t

If

with

decreases.

is an

while

if

> defeet(D').

D

is f i n i t e l y

generated

over

group.

H.

of

(a) = ( c ) .

w ~ wH

s e n e r a t e d as a b e l i a n

decompositions

Proof.

for by

are e q u i v a l e n t .

property.)

such t h a t

size

that

defect(D)

[79]).

is f i n i t e l y

is a c c e s s i b l e

so the

Thus we h a v e then

expansion

(Dunwoody

(a)

or

is p r o p e r ;

expansion

5.12 THEOREM

t h a n this,

recall

is a p r o p e F

improper

is a p r o p e r

kernel,

so may be r e p l a c e d

Let us say that the

smaller

D °gW GW W(gwGwg )

nonzero

step

gw = 1,

D'

are

map

is s u r j e c t i v e

v

wE W v

over

(a) h o l d s

qonempty H

of

has ~

family

a terminal

is a p r o p e r

t h e n we are

in the

~

of f i n i t a r y

element,

that

expansion

of it.

situation

of

g7 ACCESSIBILITY 5.11 and may, by the well-ordering, so as to minimize

defect(D).

expansion belonging to (c) = ( b ) .

choose a d e c o m p o s i t i o n

~. in particular,

family of all finitary decompositions

(b) ~ ( a ) .

If

so

G

G

is finitely generated as left as left

accessible

over

containin$

H.

Proof.

accessible

of

over

REMARKS.

have bypassed area.

H

G then

G K.

over

G

the (nonempty)

so

D~(G,~[G])

D~(G)

is

of 5.12.

over

H

and

over any subgroup

K

family of all finitary

has a terminal element so

By taking elementary some exact sequences

G

(Gv, qe I v e V,

E" : E - E'

proofs at every stage,

is

and

in their own right,

e e E) T

of

G

over

for its maximal

For any right

EndR[G](M)-linear

we

that are normally used in this

but without much detail.

for the underlying graph,

that the right

by 5.6,

D

mention them briefly,

E' : E(T),

has a

0

application

Since these are of interest

decomposition

H

H.

then,

is accessible

K

(nonempty)

over

over

H

~-module.

the

is finitely senerated

By 5.12 (b) = ( c ) ,

decompositions

5.14

If

over

G

~[G]-module,

Let us record an interesting

5.13 COROLLARY.

of

is accessible

is accessible

finitely generated

D in

Then by 5.11, D has no proper

If (c) holds then,

terminal element,

§5

map

we now

Fix a H,

write

subtree.

R[G]-module

M,

Y Set

1.5.2 says

98

III

DECOMPOSITIONS

V d is the k e r n e l

(21)

~

( (qe)d,

of a right

where

(d v)

we u n d e r s t a n d

v 0 e V,

there

is

m

)

again

Hence

we h a v e

0

M Ge

+

II D+(G ,M) E ~e e

for

~,

+ ad meq ~ - ( q e ) d r e q e I ) )

e e E'

Observe

)

~

( (mle m

= 0

e

for

it follows

is s u r j e c t i v e . and the k e r n e l also that if Thus

that

M E'' ×

of

(12)

can be d e r i v e d

that

arrow

4.12

easily

N InD(Ge,M) lies in the E M = RIG] and all the G e

if the

decomposition

v0

=

now

0.

InD(Gv,M)

which follows

f r o m this of

finite

is f i n i t a r y

from

by the

snake

Notice

fact.)

(21). then

So, (21)

÷

÷

comes

be computed.

image are

~

~ InD(Ge,M) E

can a c t u a l l y

fairly

m

[ v-{v°}

~

vertical

(Notice

and

diagram

ME

the

isomorphism

e e E'

E

and

for any

+ meq ~ - m eq ~ ) ),

M E'' × MV-{v0 } ---+

------+

that

ME

a commutative

M Gv--+

map

EndR[G](M)-linear

we u n d e r s t a n d

°}$

÷

0

(m v)

)

~+ ( ( dle = 0

e

M E'' x M V-{v0}

where

,

a right

( (me),

(diG v)

EndR[G](M)-linear

M E'' x II DH(Gv,M) V v ((me),

v

This

0

0

(21) lemma, that

5.5

shows

for e x a m p l e , is s u r j e c t i v e .

we get an exact

sequence

99

§5

ACCESSIBILITY

0

+

D~(G,R[G])

Taking

R = ~

Tor

Since

m n ~ [ G ] OGe

a short

÷

exact

~ @~[G]

finite such

that

over

H,

H.

Then

stabilizers,

we get

an exact

0.

sequence

zzE" @ @ DH(G + v) ÷ @ D H (G e ) + 0 • V v E e term

@ D~(G e)~ E e is the

vanishes,

÷

so we have

0.

problem

of u n i q u e n e s s .

7.6 of S c o t t - W a l l

is f i n i t e l y there

÷

groups

Lemma

G

+ @ R[G]/R[G]OGe E

_

Tor

section

extends

Suppose

over

e dse

the

of a b e l i a n

of this

result

THEOREM.

accessible

mOGe ,

77E'' @ @ DH( G v ) ÷ V v

topic

following

5.15

and

÷

final

=

sequence

DH(G)

Our The

and a p p l y i n g

[G](zZ,77[G]/ZZ[G ]°Ge ) + D H (G) ÷

E

0

÷ R[G] E'' @ @ D ~ ( G v , R [ G ] ) V v

exists

and h a v i n $

generated

over

a reduced

a vertex

[79]. H

and

G-tree

vH

X

is

with

stabilized

by

H,

G

is f i n i t e l y g e n e r a t e d over H and i n d e c o m p o s a b l e vH ..... and for e a c h v E V(X) - VHG , G v is f i n i t e l y s e n e r a t e d

indecomposable. For any s u c h

determined Proof.

as

Since

finitely

G of

H.

Let

X/G

is finite,

Now

is finite,

so

X

G

be the

over and

and

H

over

such

H,

that

indeeomposable,

corresponding

has

all the

let

X'

be any

desired other

as

there

E(X)

exists

vertex

over

its

right

G-tree,

are

uniquely

G-set.

each

so it can be c o n t r a c t e d

then

V(X),

is d e t e r m i n e d

is a c c e s s i b l e

generated, X

X/G

G-sets,

decomposition

which

tree,

a finitary

group

is

intersection as in 4.7.

to a r e d u c e d

G-tree,

properties.

such

G-tree.

Extend

vH

to a

with Then

100

III

OECOMPOSITIONS

connected v ¢ V, Hv,

Gv

transversal is finitely

so by 5.1,

to a m o r p h i s m exists V(X)

V

Gv

of

a morphism

= V(X'),

generated

stabilizes

G-sets of

E(X)

in V(X)

e:V(X)

G-sets = E(X')

for the over

Hv~

a vertex

v

of

X'.

D

over

This extends

and by symmetry~

÷ V(X).

G-sets.

For each

and i n d e c o m p o s a b l e

÷ V(X');

8:V(X') as

G-action.

Now by 1.6,

there

CHAPTER

IV

COHOMOLOGICAL DIMENSION ONE

With his conjecture that every torsion-free group with a free subgroup of finite index is necessarily free,

Serre essentially

initiated the study of groups of cohomological dimension one.

He

showed that any such group has cohomological dimension at most one over

~;

subsequently,

Stallings [68] showed that every finitely

generated group of eohomological dimension at most one over free,

~

is

and Swan [69] then eliminated the finitely generated part

of the hypothesis,

thus proving the conjecture of Serre.

Karrass-Pietrowski-Solitar [73],

Cohen [73],

and Scott [74]

progressively extended the group theoretical part,

concluding with

the characterization of groups which have a free subgroup of finite index,

as the fundamental groups of connected graphs of finite

groups of bounded order.

Returning to the cohomological aspect,

Dunwoody [79] more generally characterized the groups which have cohomological dimension at most one over a given arbitrary nonzero ring,

as the fundamental groups of connected graphs of finite

groups having order invertible in the given ring.

This deep result

will now be obtained fairly painlessly from III.4.6. prove a new, slightly stronger result,

In fact,

we

characterizing the

transitive group actions that give rise to a projective augmentation module. Throughout this chapter, G,

and

R

a nonzero ring.

let

G

be a g~oup,

H

a subgroup of

102

IV

COHOMOLOGICAL

I.

PROJECTIVE

AUGMENTATION

z

be an element of

We are interested R[G]-projective;

let

U

be a nonempty right

in determining

mR(U)

when

is

mR(H\G)

Gz-derivation

R -l

write

IYI • R -i if

M

is

If

N

Y

to

in the next two lemmas.

If

in

R;

thus,

integers

whose

for any set

Y,

is finite and its order is invertible M

is a risht

if and only if (iii)

is a projective

R[H]-module,

N®R[H]R[G]

For any t r a n s v e r s a l

R[U]

=

(iv)

R[U] If

@ R@R[Gs]R[G] seS

G

is

risht

R[G]-module,

we in

R.

then

(vi)

If U,

mR(U)

as

z

then

then

N

is

S

in

U

for the

G-action,

R[G]-module. if and only if

mR(U)

=

R[U-{z}],

if and 0nly if IGul e R -I for all is

IGxy I • R "I,

R[G]-projective, where

R[H]-projective

i_~s R[G]-projective.

R[G]-pr0jective

fixes

R[G]-~rojeetive

in

mR(U)

R[H]-projective.

(ii)

(v)

together

are invertible

(i)

for

We shall need some

for the set of those positive

in

LEMMA.

g ~ z - zg.

collected

images

1.1

R

conditions

The main idea is to apply III.4.6 to the

G + WR(U),

facts,

We write

for further

see Wall [71].

R[G]-projective.

elementary

the

and we shall restrict

We begin by looking at some necessary be

At present,

to proving only what is needed later;

information,

is

more generally,

R[G]-projeetive.

known results are rather fragmentary, ourselves

G-set,

U.

this section considers,

situations where

ONE

MODULES

Throughout this section, and

OIMENSION

IGul e R ~

for all ueU.

and this is u e U-{z}.

then for any distinct

Gxy = G x n Gy.

x,y

103

PROJECTIVE Proof.

(i) follows

free as right (ii).

If

from the fact that

is

R[H]-projective,

R[G]-projective.

Conversely,

R[H]-projective

R[H]-summands

(iii) follows

N

is

R

R[Gu]-projective

to

R[G]-projective

R[G] G

R[G] G

has an

=

R[U]

is

(v).

Suppose

for all

R[U]

:

R[{z}]

isomorphic

is

to

so

(vi).

Suppose

x E U,

mR(U)

R[U-{z}], to

R[G]-projective

so are all the =

=

@ N@R[H]R[HgH], H\G/H

@ R[GsXG]. seS

R[G]-projective

u e U.

if and only if

Now observe that

left inverse, p :

Rn G

but,

n G : (OG)e G.

y • U-{x}.

of cardinals,

whence

(iv).

R[G]-decomposition R[U] ÷ R[{z}] of

mR(U) ,

is

the kernel

(v).

R[G]-projective. by (i),

But

IGI e R "l ,

This proves

is R [ G x ] - p r o j e c t i v e

and

Gx

Then for each fixes

x,

so by

[] we write

largest integer that divides every finite has no finite elements.

where

by definition

is

map

(p)e G : 1.

so the kernel of

is right

IGxy I • R "I for all

with

R

which is equivalent

Then there is an

mR(U).

mR(U)

~

R[U]

if and only if

z.

@ R[U-{z}],

For any family

N@R[H]R[G]

(R[G]G)e G

fixes

is also isomorphic

is

is clearly

if and only if the augmentation

R[G]-projeetive G

N@R[H]R[G]

and hence,

is

R[G]-linear

R[{OG}],

R

@ R[gH] G/H

N@R[H]R[G]

since

containing an element

Thus

(v),

if

by (i),

from III.4.11,

By (ii) and (iii),

eG:R[G] ÷ R

=

R[H]-projective.

(iv).

right

RIG]

then

in the d e c o m p o s i t i o n

so in particular,

is

§I

MODULES

R[H]-modu!e.

N

then it is

AUGMENTATION

Y

HCF Y YeT in T,

to denote the or

0

if

104

COHOMOLOGICAL

IV

1.2

LEMMA.

(a)

The

The f o l l o w i n $

Gz-derivation

0 ÷ mR(U)

(c)

HCF (G:G u) e R -I. ueU 1 e (R[u]G)e U.

(d) If -

G

-

further (e)

equivalent G

~ R ÷ 0

is

g ~ z - zg,

is inner.

R[G]-split.

for all distinct

x,y c U

then these

are

to

stabilizes

an element

of

U,

or

G

is finite

and

(G:G u) e R a.

Proof. d

is finite

xy

Either

HCF ueU

+ R[U]

ONE

are equivalent.

dz:G ÷ mR(U), sU

(b)

OIMENSION

( a ) ~ (d).

We have

the following

equivalences:

is inner

Z

there

exists

w e mR(U)

such that

w-wg

there

exists

w e mR(U)

such that

z -w

= z - zg

for all

g e G

e R[U] G

1 e (R[u]G)su . ( b ) ~ (d) is clear. (e)~(d). where

R[U] G

=

R[{~uG [ u E U}],

the summation

principal by the

ideal

image

Finally, (c) = ( e ) ,

of

if

G

for if

(G:G u)

xy G

this

(G:G u)

so

G u n Gug

Thus

fixes

u,

in

R,

(R[U]G)~U uG.

(e) holds.

~

ideal

is principal,

which

proves

and

(c) holds,

so for any

is infinite,

= Z RIuGI,

Since

for all distinct

is infinite

and

orbits

latter

is finite

is finite,

finite, G

domain,

HCF ueU is obvious.

(e) ~ ( e )

u e U,

is over finite

so

g e G,

D

generated

(c)~(d).

x,y

in

U

then

then for some

(G:G u n Gug)

which means

is a

u = ug,

is so

g e G u.

105

PROJECTIVE AUGMENTATION IDEALS

1.3

THEOREM.

If

mR(U)

~ e n e r a t e d over

Gz,

(Gv I v ~ V)

G

o_~f

fixes

an element

Proof.

By 1.1

1.4

over of

: of

(G:G u)

Proof.

=

If

(C) ~ (e). x,y E U

the

then

G

xy G

= 1

is infinite,

in different

I

impossible. acts

orbits,

if

G

fixes

x.

on

+ (G - (G:Gy))

U - xG,

If

R[Gu]-projectiye

Proof.

follows.

ensures

Gv

e R "l.

x,y ¢ U, D

that each

Gv

G

of

x,y e U, U.

from 1.2,

Z

~

(IGzI-

so

HCF ueU

For any

Gy ~ 1

)

+

then

Z

(IGzl-

)

zcyG

½1GI

that there

since here

is finite. and

and

+ ½1G I

=

is an orbit

IGI, xG

(G:G u) = (G:Gx),

which

is

such that

G

so

G = G x,

and

0

R[G]-projective.

LEMMA.

HCF (Gv:Gvu) ueU

zcxG

We now turn to sufficient

I. 5

result

Gx ~ 1

=

It follows

freely

either

for all distinct

follows

that

u

(G - (G:Gx))

and

an element

this

zexGuyG =

v ( V,

for all distinct

fixes

Thus we may assume

IG-{ }I

for every

the result

following

is finitely

decomposition

is finite

(a) ~ (e),

G

a finitary

is finite

xy

~

If

1,

G

G

and

U.

PROPOSITION.

HCF ucU

such that

o_~r G v

1.2

R

an element

Gz

U,

and

In the case fixes

t h e n ther e exists

(vi),

so by III.4.6,

i_~s R[G]-projective,

§1

Choose

conditions

for

mR(U)

to be

Here the key step is the following.

M

is an

for all

R[G]-summand u e U,

a transversal

then

S

in

M

U

of

R[U],

and

M

is

i_~s R[G]-projective.

for the

G-action,

and for

106

IV

COHOMOLOGICAL

each

s e S,

Then

ms

write

is fixed by

R [ G ] - l i n e a r map ms@R[Gs]l. s

to

ms,

R[U]+

to an

@ M@R[Gs]R[G] seS R[U]+

s

within

of

sending each

s

M.

WR(U)

is

Thus

M

is

@ M@R[Gs]R[G], which is seS Hence M is R[G]-projective.

is an

for all

R[G]-sununand of

R[Gu]-projective

R[G]-projeetive.

to sends each

HCF(G:G u) e R -I, and IGxy [ e R -I ueU then ~R(U) is R[G]-projective. ~R(U)

R[U].

there exists an

If

x,y E U,

by 1.1 (v), is

of

@ M@R[Gs]R[G] ÷ M seS

by our hypotheses.

By 1.2 (e) = (b),

WR(U)

M-component

By 1.1 (iii),

R[G]-summand

COROLLARY.

Proof.

G s.

ONE

so is the given projection onto

R[G]-projective

distinct

for the

The composite

isomorphic

1.6

ms

DIMENSION

for all

u E U,

R[U],

so by 1.5,

D

We conclude this section with the most general conditions which 1.7

WR(U)

U,

U

Let

1.2 (c) ~ (d), such that v

= v.

moreover,

V(X)

Then V

as

WR(U)

IGxy I e R -I for all distinct

G-set, is

(Xv)g U

=

1,

for each

in

V(X)

maps.

Thus,

WR(U)

for the

v e V, if

v E U,

is an

R[U],

in such for

G-action.

an element

R[G]-linear map

this map is the identity on

augmentation

X

and

and clearly,

This then extends to an

G-tree

x,y

HCF (Gv:Gvu) e R'* uEU R[G]-projective.

be a transversal there is,

under

R[G]-projective.

further that there exists a risht

embeds in

v E V(X).

Proof.

x

Suppose that

and suppose

that all

is known to be

THEOREM.

and

By

x v E R[U] Gv

we may take R[V(X)] ÷ R[U];

and commutes with the

R[G]-summand

of

~R(V(X)).

D

107

PAIRS OF GROUPS By 1.1.1,

mR(V(X))

= R[E(X)]

suffices to show that e e E(X).

2.

v E V(X), g

R[G]-modules,

is right

e

~ g

mR(U)

is

which gives the desired result, for all

le

so by 1.5,

R[Ge]-projective

But it is clear from 1.6 that

for all since

mR(U)

as

§2

e E E(X).

it

for all

R[Gv]-projeetive by I.I (i),

[]

PAIRS OF GROUPS If

mR(G)

cohomolo~ical

is right

R[G]-projective,

one says that

dimension

at mgst one over

R,

and writes

We shall be concerned with the situation where R[G]-projective,

which includes

G

mR(H\G)

has CdRG ~ 1.

is

CdRG ~ 1 as the special case

H = 1. (For the general defintion Gruenberg

[70],

or

o f cohomological

Cohen [72].

dimension

cf

We remark that the groups of

cohomological

dimension precisely one are the infinite

cohomological

dimension

groups of

at most one.)

Our main result chacterizes

~R(H\G)

being

R[G]-projective

by

the following concept. 2.1

DEFINITION.

Let us say that

there exists a d e c o m p o s i t i o n over

H,

H

(Gv' qe

is an

R1-vertex of

I v E V(Y),

e E E(Y)

G )

if of

G

such that the following hold:

(1)

For each

e ¢ E(Y),

G

(2)

For each

v E V(Y),

either

(3)

The following integers

]HoHgl,

g

is finite;

e

Gv

are invertib!e

~ G-H;

HCF ( G v : G v n H g ) , geG

is finite,

v ~ V(Y).

in

or R:

G v = H;

108

IV

COHOMOLOGICAL

(We shall see that if satisfied that if H n Hg

H

is infinite GVH n G~H

GVH =

for if

H

H;

notice

We call a

2.2 G

e

(2).)

so for all

Notice

g• G-H,

We shall u s u a l l y

assume

is no loss of g e n e r a l i t y

in this,

G1e

= G e = H~

and take

~e

to be the

G

satisfied,

a vertex

G;

here the c o n d i t i o n

so this coincides

w i t h the d e f i n i t i o n

[73].

of

0

with

H

being an

of

R~-vertex

~R(H\G)

of

G,

being

let us note the

case.

H H F

Proof.

(3) is

vertex.

PROPOSITION. :

(1) and

going on to prove the e q u i v a l e n c e

R[G]-projective

then

is finite by (2), and we can vH with new initial vertex, and h a v i n g tel~minal

given by Cohen

Before

G

G

~'*-vertex of

(3) is t r i v i a l l y of vertex

GVH = H,

and we then set

new distinguished

satisfying

that there

is finite then

VH,

extreme

then

ONE

R'1-vertex of

is finite by II.3.1.

adjoin a new edge vertex

is an

in any d e c o m p o s i t i o n

=

that

H

DIMENSION

is a

G

=

{qel e c E}.

vertex,

v,

for all

e • E

isomorphic

by II.3.1.

Let

defines G.

V(Y),

Y set

that

suppose

F,

E•

Setting

Gv

a graph of groups whose

e e E(Y)

)

say

F

=

H

is free one

and

fundamental

Ge

=

1

group is

is a d e c o m p o s i t i o n of

for all

H

if

be the graph with e x a c t l y

It is n o w c l e a r that

Conversely,

if and only

for a free group

Thus there

It follows

G

F.

H H F

and w i t h edge

to

(Gv' qe I v e groups.

~-1-vertex of

for a free group

Suppose

on a set

H

is a

G

over

H

with trivial

g ~ G - H,

H n Hg

H

~-vertex

is a

~e-vertex

of

=

G,

edge

G v n Ggv of

G.

and let

=

I

109

PAIRS

(Gv' qe G

I v E V(Y),

= H.

So

e e E(Y)

H n Hg

vH each

v ~ V(Y),

there

is a unique

=

1

Gv g Hg

s t a n d a r d right

H-orbit

Thus,

the

Y

e e E(Y),

2.3

).

X

be the

of

V(X);

moreover,

G-orbit

for the

in

of

in G

X over

H n H qe

that

G

of pairs

: =

qe"

in for H

that we may assume

and that the d e c o m p o s i t i o n Now since

X.

H-action

transversal

It follows

are in the

1

is of

for all

H ~ F,

where

D

(G,H)

such that

we see that s u f f i c i e n c y

is

1.7.

If

THEOREM.

R[G]-projeetive.

H

is an

R-l-vertex of

p r o b l e m with the f o l l o w i n g

observation.

2.4

is

If G

G

then

WR(HXG)

is

s t a r t i n g by r e d u c i n g

the

0

We now p r o c e e d to show the converse,

R-l-vertex of

Let

decomposition

freely g e n e r a t e d by the

R[G]-projective,

G v ~ 1,

and two elements

transversal

to the c h a r a c t e r i z a t i o n

from

LEMMA.

if

for

C o n s i d e r the set

X,

we see from this p r e s e n t a t i o n

is

with

by 1.4,

G v.

H-subset

v e V(Y).

one vertex,

(H, qe I e E E(Y)

Returning

immediate

of

a (new)

for all

has e x a c t l y

is the free group

~R(H\G)

containing

is an

a connected

This gives

Gv g H

form

This

and, Thus,

and e x t e n d it to a c o n n e c t e d

G-action.

that

H

set of a subtree

we may choose

for w h i c h

of

g e G.

as in 2.1,

if and only if they are in the same

the subtree, the

g~ G-H,

as in III.4.7.

{ r e V(X) I 1 ~ G v g H}.

same

for all

conjugate

§2

be ~ d e c o m p o s i t i o n

for some

G-tree,

it is the vertex

)

OF GROUPS

WR(H\G)

if and only

R[G]-projective

then

if it is a vertex of

H G.

is an

F

110 IV

COHOMOLOGICAL

Proof.

One d i r e c t i o n

see from 1.1 remains of

G

(vi)

over

part

H

is trivial.

that

to consider

OIMENSION

In the opposite

IHn Hgl e R -I for all

a decomposition

satisfying

of (3) holds.

Let

(1) and

(2),

v c V(Y).

is inner.

we need

by (2),

R[Gv]-module HCF geG

and here

The p r o b l e m

Thus

that remains then

result

from the

suitable 2.5

comes

for getting

LEMMA.

over

H.

projective If kernel K

If

K

projective derivation there

of

exists

Gv

from

G

)

g ~ H(1 - g ) ,

if

of G.

is

v

D

~R(HXG)

G.

is

We begin with

generated

and

G

to a projective

R-Lvertex

of

is

(a) ~ ( c ) ,

case,

in the general L

and is

is finitely

is finitely

a

case.

of a derivation K

~R(HXG)

from

G

and

to a projective

generated

to a

generated

i__ss R[G]-projective,

is the kernel

L

a finitary

generated

of a derivation

then by 1.1

to a projective

as one of the vertex

finitely

then

e • E(Y)

L

over also

H. is the

R[G]-module,

then

L.

R[G]-module, from

finitely

~R(H\G)

of a derivation

is a vertex

is a vertex

H s K s L s G,

R[G]-module,

from

is to show that

is the kernel

moreover,

Proof.

K

K

(i),

So, by 1.2

is an

information

Suppose If

H

H

and it

only the case where

any derivation

R[G]-projective, that

consider

we

and show that the second

G v ~ ~R(H\G),

is inner by 111.4.2.)

(G :G n H g) e R "I. v v

g e G-H,

By 1.1

so the derivation

finite,

direction,

(Gv, qe I v • V(Y),

R[Gv]-projective, (For,

ONE

over

decomposition groups, H.

say

(i),

K

from

to a

is the kernel

RILl-module. (L w I w e W) LWK

G

So, of

By III.3

of a

by 111.4.6, L,

3,

having K

is

111

PAIRS

Now

suppose

further

R[L]-projective, 1.3,

there

such of

that X,

and

exists H

prove

that

each

we W-{WK},

vertex

by

that

of

Hg

X,

for

some

f o r all

g c G.

and this

is

Since

L

n Hg

derivation

X

with

is

or

L

~ Hg

finite of

action

finite;

L, of

it Lw

a vertex and of

it s u f f i c e s

If

ge

L,

~

then

II.3.1.

L n L g,

L

show

such

Lw

To

that

X

for

has can

groups,

v

all

be

which

would

L. act

on

is e i t h e r

so

vertex

g c G.

L-tree

finite

Lw

g c G,

some to

By

stabilizers,

for e a c h

for

each

with

let v

and

on the

of

edge

suffices

for t h e n ,

to be r e p l a c e d

by

v

hence

L ÷ WR(H\G).

finite X,

stabilizer

finite

R[G]-projective,

of

wc W-{WK},

the

is

vH

is i n d e e d

let

§2

a vertex

the

111.12

the

L-tree

is a v e r t e x

K

Thus, v

v

stabilizers

expanded show

K

L

mR(H\G)

consider

an

fixes

either

that

OF G R O U P S

to

This

finite

show

n Hg

w

~

L

w

the

to s h o w

For or

that

leaves

it s u f f i c e s

X.

each

contained

L n Hg n Kg

is

=

case

that

vertex

L

w

finite

n Lg , wK

where

for

in

g ~ G - L.

g E G - L,

w

Ln Lg

is

finite.

the

kernel

Now

for any

(gx)d

x,y

element this

of a d e r i v a t i o n x e L n L g,

= (yg)d,

since

For this,

so

e L = ker

(g)d

to h a p p e n

d.

d

say

(g)d.x

we

from gx

This

of

a projective

is

for

G

= yg,

+ (x)d

L n Lg

invoke

: says

the to

a projective

with

x , y c L,

(y)d.g that

+ (g)d, L n Lg

R[G]-module, to

be

hypothesis

finite,

as

L

is

R[G]-module.

we h a v e so

fixes

and

that

the

(g)d.x the only

desired.

= (g)d

nonzero way

for

112

COHOMOLOGICAL

IV (We remark claimed,

that the

but in its present

The m e t h o d arguments

countably

generated

(Cohen

(G8)1~8~ Y

form,

each

o < y,

8~Y,

G8 =

G

2.5 is ideal

chain T8

Let

Then

is a right

if

vI

lemmas

is based on of

just to get to the

8 = e+l

Gs-graphs,

be an ordinal,

G +1,

of

G,

induction,

such that, and there

stabilizer

such that

for

ordinal

Gy.

by transfinite

Gs-tree,

of

and suppose

and for each limit

is a vertex

of trees,

with

finite

y

of

G1

(TB)ls8~ Y

have

ease

in turn based on arguments

chain of subsroups

construct,

vertex

is

for our purposes.)

from 2.5 to the general

is a vertex

We shall

more than

case.

[73]).

u G . o<8 ~

ONE

substantially

which were

is an a s c e n d i n $

ascending

proves

We shall need two more

LEMMA

Proof.

for extending

of Cohen [73],

Swan [69].

2.6

foregoing

DIMENSION

G1,

an

for each 8 ~ Y,

is a d i s t i n g u i s h e d

and all other orbits

stabilizers;

there that

is an e m b e d d i n g

T @G G8 + T B

of

the d i s t i n g u i s h e d v e r t e x ,

preserves

v1®1 + Vl; if

Define

8

is a limit

T1 =

{v I}

with trivial

Suppose

8 ~ Y

and we have

Consider

first the case where

of

GS,

there

is a

has

stabilizer

Go,

Now

ordinal,

G°

acts on

To,

Gs-tree

then

Gl-action.

defined

T°

8 = o+l.

X = X(o),

for all Since

o < 8G

is a vertex

o

such that one vertex

and all other orbits so by III.l.1,

T 8 = o<8 u To .

have

finite

we can expand

v°

stabilizers. v

in

X

to

113 PAIRS 0Y GROUPS get a new

and there

§2

Gs-tree,

TS,

having

V(T 8)

=

(V(X)

E(T 8)

=

E(X) v ( E(T )@ G G 8 ),

- v G 8) v ( V(T )® G G8 ),

is a distinguished

vertex,

vl@l ,

with stabilizer

such that all other orbits have finite stabilizers, embedding If

8

T~SG G 8 ÷ T 8

of

GI,

and there is an

GB-graphs.

is a limit ordinal,

we define

TB =

U T .

Since

~<~ G8 =

U G ,

T8

is a

Gs-tree ,

at each step in the chain, Hence GI

GI

For

H ~ K s G,

~R(H\G)

G T.

has all the desired properties.

GB

for all

mR(H\K)G

by the natural

convenience,

we summarise

2.7

Let

(i)

~R(H\G)

=

~R(G)/~R(H)G.

(iii)

~R(HkK)G

=

~R(K)G/~R(H)G.

(iv)

~R(KXL)G

=

~R(H\L)G/~R(H\K)G.

For any subset

S

{ H ( 1 - s) I s • S}

(vii)

If

~R(HkK)G

I_ff S as

for the

R[G]-submodule

~R(HkK).

of

For

H K K ~ L g G. ~R(HkK)SR[K]R[G].

(vi)

In particular,

some simple observations.

=

(v)

8 ~ Y.

image of

~R(H\K)G

(it)

are preserved

0

we write

generated

LEMMA.

T8

is a vertex of

is a vertex of

and since stabilizers

o_~f G, senerates

~mR(HXKI)

is a subset of

R[G]-module

i_ff S u H

by

mR(HXK)G

for some G

senerates

{H(i-s)IseS}

then

a__ss R[G]-module.

H g Kl g G

such that

K,

~R(H\K)G

then

H uS

then

K

m KI.

is ~enerated generate s

K.

114

IV

Proof.

(i)

as left

follows

R®R[H]R[G]

isomorphism (iii).

that

=

~R(H\K)G

Let

from

M

derivation vanishes

RIG]

:

@ R[Kg] K\G

is flat

Let

mR(H\KI)G

=

THE

senerated

Proof.

so

be the

so by

as

g E G,

we may

for e a c h

n,

= H(1-gn

be a right

the p o s i t i v e F + ~R(H\G). for e a c h

functional,

integers, Let

and

vanishes

so

)

If

on

H u S,

so

K(1 - g)

: 0

in

~R(H\G) then

a generating

R[G]-module so there

that

Then

by

(v),

be a left

set

H

is a v e r t e x

of the

yl,Y2,..,

of

form where

gn E G. is free

(W)%n

on a set i n d e x e d

by

surjeotion

inverse

%n:~R(H\G)

w e mR(H\G),

i$ c g u n t a b l y

elements

is a n a t u r a l

n,

S u H.

step.

by the

for some

integer

by

the key

R[G]-module,

choose

for each

the

K = KI.

CASE.

~ : (~n)

positive

(vi),

is g e n e r a t e d

H ( 1 - g),

F

+ M,

generated

to prove

GENERATED

~R(H\G)

Yn

and c o n v e r s e l y ,

by

~ M.

subgroup

and p r o j e c t i v e

generated

Kl ~ K.

in a p o s i t i o n

Since

an

follows.

~R(HXG)

H(1 - g) e ~ R ( H \ K ) G ,

g e K,

COUNTABLY

induces

(~R(K)/~R(H)K)@R[K]R[G].

of

g ~ H(1 - g)

WR(H\K)G

mR(HkK)G,

We are now

(ii)

=

M ~ ~R(H\K)G,

g • KI,

KI

and

R[G]-submodule

Then

Thus

and this

(iii).

so

For any

(vii).

R[H\G],

~R(H\K)@R[K]R[G]

be the

K,

~R(K\G).

is,

ONE

by I I I . 4 . 1 1 ,

=

G ÷ ~R(H\G)/M,

on

(vi).

Let

fact

R[HXG]

=

{H(1 - s) I s • S}.

2.8

from the

R[G]/~R(H)G

is c l e a r

(v).

OIMENSION

R[K]-module.

(ii).

(iv)

COHOMOLOGICAL

of this + R[G]

map;

that

is a

= 0 for almost

all

n,

G.

115

PAIRS OF GROUPS

and

w

=

~ yn.(W)%n. (Thus we are c h o o s i n g a " p r o j e c t i v e n~l system" for mR(H\G).) For each n ~ 1, we have a

coordinate derivation H(1-g)

dn:G + R[G], :

that

Gm

R[G]-module, U m~l

g ~

C H(1 - g)

~ yn. Cg)dnn~l positive integer

For each Notice

§2

m,

is the k e r n e l

g ~

( gdn

= G. m For each p o s i t i v e

let

)%n'

for e a c h

G m = {geG I gdn=0

of a d e r i v a t i o n

)n~m"

and

Also

from

g e G,

for all n~m}.

G

to a free

H = G 1 ~ G 2 ~ ...,

and

G

generated

by

part

course part

of 2.5,

Gm+ 1

To pass result,

2.9

to the

y,

that

general

Y

6 = ~+1

of

[58]).

then

generated over

Gm+l,

Let

if PB/P

set

chain

Py = P, and

of

H.

Thus

Hence

by the

H.

So of

over

Now by the

and t h e n

G

second

by 2.6,

H : G1

D

any ~enerating

Y,

subgroup

on the

following

classical

for our purposes.

and an a s e e n d i n ~

of

be the

Gm -< L m .

case we rely

modified

P1 = O,

by a subset

generated

as claimed.

(Kaplansky

and

ordinal

and if

G,

Lm

so { y i , . . . , y m } ~ R ( H \ L m ) .

is f i n i t e l y

is a v e r t e x

slightly

R-module,

such

Gm of

THEOREM

Gm

let

2 . 7 (vi),

so by

is f i n i t e l y

of 2.5,

is a v e r t e x

m,

H u { g l , . . . , g m },

=_ ~ R ( H \ L m ) G ,

~R(H\Gm) first

integer

and 6

P

be a p r o j e c t i v e

for

P.

Then

(P6)l~y

there

exists

of s u b m o d u l e s

for each

6 ~ Y,

is a limit

ordinal

is a c o u n t a b l y

right

P6 then

~enerated

an

of

P

is ~ e n e r a t e d P6 =

u

P ,

projective

R-module. Proof. ~y:P + R,

As

in the y e Y,

preceding such

that

proof, for each

there p e P,

exist

functionals

(p)~y

= 0

for almost

116

COHOMOLOGICAL

IV

all

y E Y, Let

Z

y E Y -A,

p

be the we h a v e

unions. Z

and

Now set S,

that

A

Define

:

A

is f i n i t e

Y1

Y,

Y S

we

s u c h that is c l o s e d

construct

~ 0

for some

for all

under

Y

for all

and we

Y

we d e f i n e

Y8

=

u

Y

eventually

gives

For e a c h

we

arbitrary

that

u ~. .

Since y

set

is p r o j e c t i v e .

P,

retraction P

A

chain

in

8 Y

define

of

P

is a set

has

P ~

that Z

is a s u m m a n d

S

as

then

follows.

either

element

,

YyR

Notice

p • P - Y R,

ordinal, this

process

= P.

all the d e s i r e d is t h a t P

for e a c h

is a d i r e c t

Y'(P)¢Y" P~+I'

properties.

Hence which

~ < y, summand

each

completes

Q

We n o w a r r i v e hypothesis

that

on w o r k

at the m a i n CdRG

~ I,

result. this was

of S t a l l i n g s ,

Swan,

then

We c l a i m t h a t the

immediate

Observe

some

belongs

is c o u n t a b l e .

is a l i m i t

such t h a t

of

and we h a v e

8 = ~+1,

P8 = Y8 R"

P ÷ Pc'

so

exists

If

is not

Pe+I/P

If

A

a e An}.

is an o r d i n a l

there

of s u b m o d u l e s

The o n l y p r o p e r t y

and

B

or

an o r d i n a l

8 ~ Y,

chain

projective,

an a s c e n d i n g

~ < 8.

stop,

An,

a e A,

an e l e m e n t

and i n d u c t i v e l y

so is e a c h

Now suppose

=

building

of

A 0 = A,

then

Y8

with

A

construct

= ¢"

P,

resulting

Clearly

{ y c Y [ (a)$y

and we d e f i n e

proof.

0.

of

u A . It is f a i r l y e a s y to see t h a t A n~0 n is the s m a l l e s t e l e m e n t of Z containing A.

constructed Y R

:

inductively

We set

=

A

=

and

if

We

set of s u b s e t s

For any s u b s e t

An+ 1

ONE

Z y.(p)¢y. yeY

(a)¢y

as follows.

to

=

DIMENSION

W i t h the proved Cohen

additional

by D u n w o o d y et al.

[79],

of

P~ the

is

117

PAIRS OF GROUPS

2.10 an

THEOREM.

mR(HXG)

R'Z-vertex of

Proof.

is

R[G]-projective

to show that if

R[G]-projective

then

H

generating

Y

{H(1 -g) I g~ G},

a chain of we have, of

G

B

=

is a vertex of

R[G]-submodules for each

containing

subgroups

of G,

of

6 s Y, H.

=

~R(G \GB)G

=

mR(G \Gs)SR[GB]R[G]

projective

R[Gs]-module , generated.

conjecture

2.11

G

{ H ( l - q e ) I e ~ E}.

by

is then a

seen that it must be

is a vertex of

and proved by Dunwoody

for a free group

is freely generated

G~

generated

GB,

2.10 and 2.3 give an equivalence

is

then

so by

D

is finitely generated m77(H\G)

mR(Ge\G 8)

and it is easily

by Wall [71],

THEOREM.

G : H HF

R = ~,

B = ~+1,

by 2.7 (i).

by 1.1 (ii),

G.

B ~ Y, if

by 2.7 (iv),

So by 2.8,

is a vertex of

In the case

where

=

G8

(GB)I~B~ Y of

and for each

=

R[G]-module;

H

G8

By 2.7 (v),

we have a chain

G 1 = H, Gy = G,

is

be

for some subgroup

u G , and if e<6 (mR(H\Gs)G)/(mR(H\G)G)

projective

2.6,

is

the

(PB)l~B~y

as in 2.9.

we are given that this is a countably

countably

H

mR(H\G)

Consider

and let

~R(H\G)

By 2.7 (vi),

such that

PB/P

G.

P8 = ~R(H\GB)G

is a limit ordinal then

From 2.9,

if and only if

G.

From 2.3 and 2.4, it remains

set

§2

over

{qeleeE}

Moreover, then

[79] in the case

H.

77[G]-projective F.

that was

if and only if in this event,

~R(H\G)

is

if

R[G]-free

F on

118

COHOMOLOGICAL

IV Proof. tree

The second part X

having

mR(V(X))

V(X)

= R[E(X)]

We now return

2.12

THEOREM

(a)

mR(G)

(b)

G

groups (c)

G

to 2.10

acts

3.

FINITE

call

an

Proof.

in

this mR(G)

in such

in

graph of finite

R.

a way that the orders

are finite

of the

and invertible

in

R. D

for abstract

of 11.3.6.

We first

group

theory

require

is the

a result;

subgroups

have

let us

order

is finite,

and

G

is an

R'1-group,

and

CdRG ~ 1. R[H]

+ mR(G)

sequence

R[H]-modules. splits,

=

0 ÷ mR(H)

Since so if

RIG],

RIG]

mR(H)

and

R[H] n mR(G)

÷ R[H]

@ mR(G)

is right is

=

mR(H),

+ RIG] ÷ 0

R[H]-projective,

R[H]-projeetive,

then

R[H]-projective.

Now c o n s i d e r right

of 2.12

an exact

sequence is

are equivalent.

of a connected

are invertible X

If (G:H)

Clearly

of right

H = 1.

R.

then

so we have

group

R'I-group if all its finite

LEMMA.

CdRH ~ 1,

the case

EXTENSIONS OF FREE GROUPS

converse

3.1

acts on a

and by 1.1.1,

The following

of the vertices

promised

invertible

G

R[G]-projective.

orders

The implication

G

E(X)

[79]).

on a tree

stabilizers

from the fact that

and record

fundamental

whose

ONE

= V I\G, E R[G]-modules. []

(Dunwoody

is the

follows

= H\G,

as

is right

DIMENSION

first

R[G]-module

M,

the case where

(G:H) e R -I.

the m u l t i p l i c a t i o n

map

Here,

M@R[H]R[G]

for any + M

has

119

FINITE

a left

inverse

M

R[H]-projective,

is

~R(G)

M ÷ MSR[H]R[G],

is right

the preceding,

Serre

argument

proved

p.9.

is

in the ease where

the analogue

in the

proof.

H-stabilizers.

It would

say,

3.2 G

finite Proof. free G

to show that if finite

edge

cd~F s 1 the

G

[73],

fundamental

groups

of b o u n d e d

One direction

freely

G

is

(G:H)

F

G/F

group order

gives

then

and G

acts

the next

generated [73],

of finite

H

of the

case of

the countably

step to Scott

of a connected

is 11.3.6.

of finite

on

is finite

illuminate

final

§1.7,

extensions

The finitely

and the

grgup

[71],

[74].

index if and only

(faithful)

sraph

of

order. Conversely,

index.

Then every

so has order

by 2.11 or 2.12,

whose

since

by

to know if Serre's

finite

would

has a free subgroup

is the

fundamental

groups,

Serre

stabilizers,

being

of 11.3.6.

ease to Cohen

subgromp

acts

so,

is commutative,

is due to K a r r a s s - P i e t r o w s k i - S o l i t a r

THEOREM.

if

R

be interesting

Such a construction

the converse

generated

cd~H s 1

of 3.1 for arbitrary

R = ~,

on some tree with the stabilizers

result

(G:H) • R "l.

CdRG s 1

case where

In the case

acts on a tree with,

this

Z m~l®g, so if HgEH\G R[G]-projective. Thus

so by 2.12 again,

could be r e f i n e d

result,

(G:H) ~

we see from 2.12 that

dimension,

[72],

a topological

M

§3

GROUPS

D

Originally,

cf Cohen

ease,

cd~G s 1,

R-Lgroup.

cohomological

then

OF FREE

m ~

R[G]-projeetive

In the general

an

EXTENSIONS

so by

3.1,

of a connected is bounded

dividing

by

(G:F).

D

finite (G:F).

cd~G s 1, faithful

suppose

G

subgroup

of

Further,

so by 2.12,

graph

has a

of finite

G

is

120

IV

COHOMOLOGICAL

DIMENSION

Let us note some consequences

of 2.12 and 3.2.

3.3

THEOREM.

(a)

e d R G ~ 1,

and the finite subgroups

(b)

G

is an

R-l-group,

(c)

G

is the fundamental

The following are equivalent.

of finite Proof.

ONE

of

G

are of bounded order.

and has a free sub~rgu P of finite index. group of a connected

(faithful)

graph

R'1-groups of bounded order.

(a)~(c)

by 2.12,

3.4

THEOREM.

(a)

C d R G ~ 1,

and

(b)

G

R-l-group,

and ( b ) ~ (c) by 3.2.

The following are equivalent.

is an

G

is finitely senerated. and has a finitely generated

free

subgroup of finite index. (c)

G

is the fundamental

graph of finite Proof.

(a)~(c)

(faithful)

R'1-groups.

by 2.12,

It is appropriate

group of a finite connected

and ( b ) ~ ( c )

by 3.2. D

that we should conclude with the

R = ~

case

of 3.3. 3.5

THEOREM

(Serre-Stallings-Swan).

The following are

equivalent. (a)

cdzzG -< 1.

(b)

G

is torsion-free,

and has a free subgroup of finite index. o

(c)

G

Proof.

is free. (a)~(c)

by 2.12,

and ( b ) ~ ( c )

by 3.2.

D

BIBLIOGRAPHY

AND

AUTHOR

INDEX

The page references at t h e e n d o f t h e e n t r i e s indicate the p l a c e s in t h e t e x t w h e r e t h e e n t r y is q u o t e d ; other references t o an a u t h o r a r e l i s t e d a f t e r h i s n a m e .

Bamford, C. and Dunwoody, M.J. 76.

On a c c e s s i b l e

groups.

J.

Pure

~ A p p l i e d Algebra

7 (1976)

333-346.

[ 88,90 Bass, H.

[vi,20,21,23,35,36,37,39,41,48,55

76.

Comm. Algebra

Some remarks on groups acting on trees. 1091-1126.

4 (1976) [27

Bergman, G.M. 68.

On groups acting on locally finite graphs. 335-340.

Ann. of Math. 88 (1968) [66

Brltton, J.L. 63.

The wordproblem.

Chlswell, 73.

Ann.of Math. 77 (1963) 16-32.

I.M.

[44 [ vi,vii,21

On groups acting on trees.

PhD Thesis, University of Michigan, Ann Arbor, Michigan, 1973. [ 20,21 76. Exact sequences associated with a graph of groups. J. Pure and Applied Algebra 8 (1976) 63-74. [20 76'. The G r u s h k o - N e ~ a n n Theorem. Proc. London Mat. Soc.(3) 33 (1976) 385-400. [50 77. Groups acting on trees. Lecture course at Queen Mary College, London, 1976/7. [21,50 79. The Bass-Serre Theorem revisited. J. Pure ~ndApplied Algebra 15 (1979) 117-123. [21 Cohen, D.E.

[vii,ll6

70. 72.

Math. Zeit. 114 (1970) 9-18. [63,66 Groups of cohomological dimension one. Lecture Notes in Mathematics,

73.

Vol. 245. Springer, Berlin, 1972. Groups with free subgroups of finite index,

Ends and free products of groups.

[vi,i07,119 pp26-44 in Conference on Group Theory, University of Wisconsin-Parkside 1972. Lecture Notes in Mathematics, Vol. 319. Springer, 1973. [58,101,108,112,119

Dicks, W. 77. 79.

Mayer-Vietoris presentations over colimits of rings. Proc. London Math. Soc.(3) 34 (1977) 557-576. [21 Hereditary group rings. J. London Math. Sod. 20 (1979) 27-39. [21

Dunwoody, M.J.

79.

see also

B a m f o r d , C.

[v,vi,87

Accessibility and groups of cohomological dimension one. Proc. London M~th. Sot.(3) 38 (1979) 193-215. [27,31,34,55,66,69,72,77,82,88,90 92,96,101,116,117,118

122

BIBLIOGRAPHY

AND AUTHOR

INDEX

Gruenberg, K.W. 70.

CohGmologicaltopics in group theory. VOI. 143.

Lecture

Notes in Mathematics,

Springer, Berlin, 1970.

[107

Grushko, I.A. 40.

Uber die Basen eines freien Produktes yon Gruppen. 169-182. (Russian. German summary.)

Mat.Sb. 8 (1940) [53

Higgins, P.J. 66.

Higman, G., 49.

[vi

Grushko's Theorem.

J.Algebra 4 (1966) 365-372.

[49

Neumann, B.H. and Neumann, H.

Embedding theorems for groups.

J.London Math.Sot.

24 (1949) 247-254. [14

Kaplansky, I. 58.

Karrass, A., 73.

Ann.of Math.

Projective modules.

[115

68 (1958) 372-377.

Pletrowski, A. and Solitar, D.

Finite and infinite cyclic extensions of free groups. Math.Soc. 16 (1973) 458-466.

J.Australian [46,101,i19

Kurosh, A.G. 37.

Zum Zerlegungsproblem der Theorie der freien Produkte. 2 (1937) 995-1OO1. (Russian. German snmmary.)

Rec.Math.Moscow [48

Lyndon, R.C. and Schupp, P.E. 77.

Combinatorial group theory. Ergebnisse der Mathematik 89.

Springer,

Berlin, 1977.

Neumann, B.H. 43.

See

Higman, G.

On the number of generators of a free product. (1943) 12-20. see also

Neumann, H. 48.

also

J.London Math.Soc. 18 [53

Higman, G.

Generalised free products with amalgamated subgroups I70 (1948) 590-625.

Amer.J.Math. [48

Passman, D.S. 77.

The algebraic structure of group rings.

Pietrowski,

32.

see

A.

Reidemeister,

Karrass,

John Wiley and Sons, London 1977.

A.

K.

Einfuhrung in die kombinatorische Topologie. (Reprinted by Chelsea, New York, 1950.)

Schupp, P.E.

S~e

vieweg, Braunschweig, 1932. [36

Lyndon, R . C .

Scott, G.P. 74.

An embedding theorem for groups with a free subgroup of finite index. Bull. London Math.Soc. 6 (1974) 304-306. [101,119

123

BIBLIOGRAPHY

AND

AUTHOR

INDEX

Scott, G.P. and Wall, C.T.C. 79.

Topological methods in group theory, pp137-203 in Homological Group Theory. LMS Lecture Notes 36. Cambridge University Press, [vi,79 Cambridge, 1979.

Schreier, 27.

Die Untergruppen der freien Gruppen. 161-183.

Serre, 71. 77.

O.

[36

J.-P.

[v,vi,35,36,55,1Ol,l19,120

Cohomologie des groupes discrets, pp77-169 in Prospect8 in Mathematics. Ann. of Math. Studies 70. Princeton, 1971. [119 Arbres, amalg~me8 et SL 2. Ast~risque No. 46. Soci~t~ Math.de France, 1977. [12,20,21,23,24,27,37,39,41,47,48

see

Solltar, D. Stalllngs, 68.

Abh. Math. Univ. Hamburg 5 (1927)

Karrass,

A.

J.R.

[v,vi,ll6,120

On torsion-free groups with infinitely many ends. (1968) 312-334.

Swan, R.G. 69.

Ann. of Math. 88 [55,85,101 [v,I16,120

Groups of cohomological dimension one.

J. Algebra 12 (1969) 585-610. [101,112

Wagner, D.H. 57.

On free products of groups.

Trans. Amer. Math. Soc. 84 (1957) 352-378. [52

Wall, C.T.C. 71.

see also

Scott, G.P.

Pairs of relative cohomological dimension one. Algebra 1 (1971) 141-154.

J. Pure and Applied [87,102,117

SUBJECT Accessible (over a subgroup),87 Act, 7,8 Act freely,36 Admit (a derivation),74 Almost contain,30,63 Almost equal,30,63 Almost-right-invariant,30,63,75 Almost right subset, 30,69 Augmentation ideal,99 Augmentation map,79 Augmentation module,79 Automorphism of graphs,4 Bouquet of loops, 36 Cayley graph,8 Chosen successors,67 Circuit, 2 Coboundary,62 Cohomological dimension (one),lO7 Colimit, 48 Component, 2 Connected, 2,6,10,63 Connecting family,9 Contract,61 Coproduct,48 Coproduct with amalgamation, 14 Cover, 32 Covering,6 Cut,62 Decomposable (over a subgroup),85 Decomposition,55 Decomposition over a subgroup,69 Defect,95 Derivation,16 Edge,l Edge group,lO,56 Expand,73,58 Expansion,95 Extremity,27 Faithful,41 Fibre bundle,58 Finitary,69 Finitely generated over a subgroup,68 Fix = stabilize,27 Forest, 2 Free action,36 Free product,48 Free product with amalgamation, 14

INDEX

FulI,66 Fundamental group,lO,40 G-action,7 G-bimodule,16 G-graph,8 G-set,7,8 G-tree,8,79 Geodesic,2 Generate finitely over a subgroup,68 Graph, l Graph automorphism,4 Graph of groups,lO Group action, 7 Group ring,15

Groups,lO H-derivation,75 HNN construction,14 HNN extenSion,15 Improper expansion,96 Inaccessible (over a subgroup),87 Indecomposable (over a subgroup),85 Induced G-set,56 Infinite path, see path Initial,l Initial extremity,27 Inner derivation,16 Interval,31 Isomorphism of graphs,4 Length, 2 Local isomorphism,5 Locally injective,5 Locally surjective,5 Loop,l Maximal subtree,3,6 Morphism of graphs,4 Outer derivation,82 Orbit,7 Path,l Path format,44 Path in universal specialization,40 Points to, points away from, 29 Proper decomposition,82 Proper expansion,96 Proper S-cut,63 Proper almost-right-invariant set,82

126

SUBJECT

R'l~roup, i18 R-l-vertex, IO7 Rank, 52 Reduced G-tree, 61 Reduced path, 2,44 Reduced form, 2 Reduction, 2,44 Right G-set, 8 Right G-tree, 79 S-coboundary, 62 S-cut,63 Shift, 28 Simple reduction, 2 Size, 94 Source, 29 Spanning tree, 3 Specialization, 38 Stabilizer, 7 Standard graph, 12,40 Standard tree, 12,40,79

INDEX

Star, 4 Structure Theorem, 23 Subgraph, 1 Terminal, 1 Terminal extremity, 27 Thin,66 Transversal, 6,7 Tree, 2 Tree of groups,48 Tree product, 48 Trivial, 35,56 Underlying path,41 Unreduced G-tree, 61 Universal covering, 6 Universal specialization, 39 Vertex,1 Vertex group, i0, 55 Vertex of a group,108

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