Combinatorial Group Theory and Topology. (AM-111), Volume 111 9781400882083

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Annals of Mathematics Studies Number 111

COMBINATORIAL GROUP THEORY AND TOPOLOGY EDITED BY

S. M. GERSTEN AND

JOHN R. STALLINGS

PRINCETON U N IV ER SITY PRESS

PRIN CETO N , N EW JE R S E Y 1987

Copyright © 1987 by Princeton University Press ALL RIGHTS RESERVED

The Annals of Mathematics Studies are edited by William Browder, Robert P. Langlands, John Milnor, and Elias M. Stein Corresponding editors: Stefan Hildebrandt, H. Blaine Lawson, Louis Nirenberg, and David Vogan

Clothbound editions of Princeton University Press books are printed on acid-free paper, and binding materials are chosen for strength and durability. Paperbacks, while satisfactory for personal collec­ tions, are not usually suitable for library rebinding

ISBN 0 -6 9 1 -0 8 4 0 9 -2 (cloth) ISBN 0 -6 9 1 -0 8 4 1 0 -6 (paper)

Printed in the United States of America by Princeton University Press, 41 William Street Princeton, New Jersey

☆ Library of Congress Cataloging in Publication data will be found on the last printed page of this book

CONTENTS PREFA C E

vii I Com binatorial Group Theory

PRO BLEM S IN COMBINATORIAL GROUP TH EO R Y by R oger Lyndon

3

POIN CARE D U A LIT Y GROUPS OF DIMENSION TWO A R E SU R FA C E GROUPS by B eno Eckmann

35

HOW TO G E N E R A L IZ E O N E-R ELA T O R GROUP TH EO R Y by Jam es Howie

53

G RAPH ICA L T H EO R Y O F AUTOMORPHISMS OF F R E E GROUPS by John R . Stallin gs

79

P E A K REDUCTION AND AUTOMORPHISMS OF F R E E GROUPS AND F R E E PRODUCTS by Donald J . C ollin s NONSINGULAR EQUATIONS OF SMALL WEIGHT OVER GROUPS by S. M. G ersten

107

121

A G R A PH -TH EO R ET IC LEMMA AND GROUP-EMBEDDINGS by John R . S tallin gs

145

T H E TO D D -C O XET ER PR O C ESS, USING GRAPHS by John R . S tallin gs and A. R o y ce Wolf

157

A SUBGROUP THEOREM FO R PREG R O U PS by F ran k Rimlinger

163

GROUPS WITH A RATIONAL CROSS-SECTION by Robert H. Gilman

175

ON T H E RATIONAL GROWTH O F V IR T U A L L Y N ILP O T E N T GROUPS by Max B enson

185

SJO G R EN ’S THEOREM FO R DIMENSION SUBGROUPS TH E M ETA BELIA N CASE by Narain Gupta

197

v

vi

CONTENTS

ON GROUP PR ESEN TA TIO N S, COPRODUCTS AND IN VERSES by Robert C raggs and Jam es Howie ON C O M PLEX ES DOMINATED B Y A TW O-COMPLEX by John G. R a tcliffe

21 3

221

SU B C O M PLEX ES OF TW O-COM PLEXES AND P R O JE C T IV E CROSSED MODULES by Michael Dyer

255

LENGTH FUNCTIONS OF GROUP ACTIONS ON A -T R E E S by Roger Alperin and Hyman B a s s

265

II Very Low Dim ensional Topology RESID UAL FIN ITEN ESS FO R 3-MANIFOLDS by John Hempel

379

T H E NIELSEN -THURSTON TH EO R Y OF SU R FA C E AUTOMORPHISMS by Steven A. B le ile r

3 97

WHITEHEAD GROUPS OF C ER TA IN H Y P E R B O L IC MANIFOLDS, II by A. J . N icas and C. W. Stark

415

A CHARACTERIZATIO N OF F IN IT E SUBGROUPS OF TH E MAPPING-CLASS GROUP by Ja n e Gilman

433

A SEQ U EN C E OF PSEUDO-ANOSOV DIFFEOMORPHISMS L . Neuwirth and N. P atterso n

443

D EH N ’S ALGORITHM R EV IS IT ED , WITH A PPLICA TIO N S TO SIM PLE CURVES ON SU R FA C ES by Joan S. Birman and C aroline S eries

451

PATHS OF GEODESICS AND GEOM ETRIC IN TERSECTIO N NUMBERS: I by Marshall Cohen and Martin L u stig

479

PATHS OF GEODESICS AND GEOM ETRIC IN TERSECTION NUMBERS: II by Martin L u stig

501

S E L E C T E D PROBLEM S by S. M. G ersten

545

PREFA C E Com binatorial Group Theory liv es in the fertile region betw een pure group theory and pure topology.

The interplay betw een the lo g ica l pre­

cisio n of algebra and the intuitive depths of geometry gives it charm and strength.

Lyndon s u g g e sts in h is a rtic le that perhaps there is no e x a c t

definition of the su b je ct.

C ertain ly it inclu d es the study of equations

over groups (which tends to involve the study of geom etric diagram s), much of 3-manifold theory (the P o in care C onjecture is equivalent to a group-theoretic q u estion ), group a ctio n s on geom etric-com binatorial o b jects such a s tre e s , the theory of su rfa ce automorphism s, and many other developm ents.

A s the Walrus said , the time has come to talk of

many things. At A lta Lodge in the sp e cta c u la r W asatch Mountains of U tah, on Ju ly 1 5 -1 8 , 1 9 8 4 , six ty of us gathered for an intense co n feren ce.

Roger

Lyndon’s opening a d d re ss, in the tradition of H ilbert, offered a s c o re of open problems to guide the field in the future.

In the s ty le of the

Seminaire Bourbaki, s ix sp eak ers were assig n ed to p ics for exp ository ta lk s ; th ese were the origins of the papers in this book by Alperin and B a s s , B le ile r, Eckm ann, Hempel, and Howie.

In addition, there were

about tw enty-five shorter talk s on current re se a rch , from which we s e le c te d the rem aining a rtic le s h ere. Our goal was to produce a book full of id e a s, understandable to our­ s e lv e s and to stu d en ts, that will open up the future developm ent of th is field .

We co n scio u sly tried to reach a large c la s s of re a d e rs, including

good graduate stu d en ts with backgrounds in group theory and topology. We are grateful for the coop eration of the authors who have helped us in pursuing this goal.

viii

PREFA C E

It is a p leasure to acknow ledge the a s s is ta n c e given us by R oger Lyndon, with whom we frequently con sulted in the planning of the co n ­ feren ce.

In tribute to his work in the field and his insight into d irection s

for future re se a rch , h is a rtic le has a prominent p lace in this volume. We are greatly indebted to Jam es Howie and Geoffrey Mess for frequent co n su ltatio n s.

In addition, we wish to thank some thirty

re fe re e s, who by tradition must remain anonymous, for their valuable aid. S p ecial thanks are due to Marty Jo n es of A lta Lodge and to Ann Reed for helping make the co n feren ce run sm oothly.

F in a lly we acknow ledge

with thanks and appreciation the fin an cial support of the U niversity of Utah and the N ational S cien ce Foundation. STEVE GERSTEN JOHN STALLINGS

Combinatorial Group Theory and Topology

PRO BLEM S IN COMBINATORIAL GROUP TH EO RY Roger Lyndon 0.

Introduction Steve G ersten ask ed me to give a talk like Hilbert gave in P a ris in

1900.

I said Pd be happy to , but pointed out th at Pm no H ilbert.

Merzlyakov [1 1 7 ] cam e to my re sc u e with the following su g g estio n : “ R ather than w aiting [for a new H ilbert] group th e o rists have com e to the more p rosaic idea of a P resen t-d ay c o lle c tiv e H ilb e rt.”

So I have

appealed to some of you for amendments to a provisional outline, for which I thank you, and I hope th is exo n erates me of any ch arge of arrogance in presenting this b iased survey. What is Com binatorial Group T heory?

T his term acquired o fficial

s ta tu s a s the title of the book of M agn us-K arrass-S olitar.

The first

se n ten ce of the History of Com binatorial Group Theory by Chandler Magnus [2 8 ] s a y s :

“ Com binatorial Group Theory may be ch a ra cte riz e d

a s the theory of groups which are given by g en erators and re la tio n s — ” (But compare M agnus-K arrass-Solitar with C oxeter-M oser, G enerators and R elation s for D iscre te G roups.) T his hardly does ju s tic e to the g oals or methods of the su b je ct.

On other o cc a sio n s Chandler-Magnus sp eak of

“ group theory with the excep tio n of L ie groups and of group re p resen ta­ tions and linear g ro u p s,’ ’ but this seem s to both include and e xclu d e too much. Maybe we could define Com binatorial Group Theory to be Very Low Dim ensional T opology.

In fa c t, group th e o rists , like oth ers, w ill a tta ck

w hatever problems in te re st them with w hatever tools they have a t hand, so perhaps Com binatorial Group Theory is ju st a s ta te of mind. Maybe it

3

4

ROGER LYNDON

is distinguished by a relu ctan ce to make the great supposedly sim plifying assum ptions of Com mutativity, L in earity , and either F in ite n e s s or Con­ tinuity, or, put more p o sitiv ely , a relish for the com binatorial co re of a problem. A ltogeth er, it seem s fru itless to try to define an elephant we have barely touched. I intended to talk about the history of the su b je ct, but ChandlerMagnus have let me off that hook.

I do not need to te ll anyone that the

major so u rce and strength of Com binatorial Group Theory has been Topology, wherein we ten tativ ely include D iscontinuous Groups, from P o in care on, with an a s s i s t from the more or le s s a b s tra c t or axiom atic sid e , beginning with C ay ley , through the influence of F in ite Groups and of problems from L o g ic .

T h ese influences will be evident, and, if my

d iscu ssio n appears biased a g a in st topology, it is b e ca u se a s a non­ top ologist I am reluctant or unable to te ll you what you know b etter than I. A ltogeth er, though, I was agreeab ly surprised by the homogeneity of the su b ject, if somewhat inconvenienced by the many interconn ections among the various ap proaches and problems in the su b je ct. My accou n t is a ls o biased for the most part toward rather recen t work, and to pathways to the future more than monuments to the p ast.

The s p a ce

devoted to a su b ject should not be taken a s a m easure of the im portance attach ed to it.

L ik e w ise , the mention of a name or cita tio n of a paper is

often more to draw attention to it than to bestow an honor.

R eferen ces

are illu strativ e or su g g e stiv e , and should usually be followed by ‘and o th e rs .’ I have not cited papers that are well known or e a s ily a c c e s s ib le (with a few e x ce p tio n s), in particular papers more than about five y ears old, or listed in the bibliography of Lyndon-Schupp.

Incomplete referen ces

in d icate lack of knowledge. D espite the unity of the su b je ct, a s a first ste p toward lin earization I have divided it into seven unequal and gerrymandered se c tio n s .

It took

some cutting and pasting to bring my lis t of problems to twenty, a modest three fewer than H ilbert. c l a s s i c a l form:

Even s o , many are really ‘ problem a r e a s ,’ of the

‘What can be said about — ? . ’

PROBLEMS IN COMBINATORIAL GROUP THEORY

1.

5

P ro p erties of fr e e gro u p s, equations in an d over groups The most ‘ a b s tr a c t’ and ‘a x io m a tic ’ of our problems is a folklore

problem of Alfred T a rsk i. PRO BLEM 1.

Do a ll n onabelian fr e e groups have the sa m e elem en ta ry th eo ry ?

The elem entary theory of a c la s s of groups is the s e t of a ll se n te n ce s in first order logic (with symbols for equality and group com position but, em p h atically, exclu d in g s e t theory) th at are true in a ll groups of the c la s s .

By co n tra st, among free a b elia n groups those of rank a t most 2

are distinguished by the property th at there e x is t elem ents a su ch th at, for a ll x , one of x , xa , xb , xab

and b

is the square y 2 of

some elem ent y . See V. Dyson [4 7 ]. In this co n n ectio n , R . Vaught posed the following te s t problem: free group, does

In a

x 2 y 2 = z 2 alw ays imply xy = yx ? T his w as proved by

Lyndon and gen eralized e x te n siv e ly .

The method w as c lo s e to that used

by H. Z iesch an g in studying automorphisms of su rface groups and F u ch sian groups. A much earlier theorem of Frob en iu s (1 8 9 5 ) d eals with the number of solu tion s of the equation x 11 = 1 in a finite group; for a survey of this problem se e H. F in k e lste in [57].

See a ls o Finkelstein-M andelberg [58].

Equations over groups entered Com binatorial Group Theory in a paper of B .H . Neumann (1 9 4 3 ), where he showed th at, for a p ositive integer n and an elem ent g of a group G , the equation x n = g has a s many so lu ­ tions as desired in som e group H con tain in g G. Higman-NeumannNeumann exten sio n s are a g en eralizatio n of the fa c t th at the equation t- 1 g1t = g 2 , for g x and g 2 in G , has a solution containing G if and only if g^

t in a group H

and g2 h ave the same order.

The ce n tra l problem in th is area is the K ervaire-Lau d erb ach problem, which we sta te as follow s.

6

ROGER LYNDON

PRO BLEM 2.

If G has a p resen ta tio n G = (X :R ) and H = (XU tiR U w )

is obtained by adding one new gen era to r an d on e d efin in g rela tio n , when d o es the in clu sio n X -^X U t in d u ce an in jectio n of G into H ? In sim pler language, for g ^ ' ^ g n w(gl>, , , >gn>t) = 1 have a solution

G , when does an equation

t in some group H containing G ?

We have seen that t~ 1g 1tg 2 ~1 = 1 has no solu tion if gj have different orders.

and g2

G erstenhaber-Rothaus showed that if the sum of

the exponents on t in w is not 0 , and if G can be embedded in a com pact con nected L ie group, then a solution alw ays e x is ts .

Rothaus

[1 44] later improved the condition on G to lo ca l residu al fin iten ess. The su fficien t condition of lo cal in d icability has been studied by J . Howie [94].

The group G = ( a 1 ,- - - ,a 4 : a j+1~ l a ja j+ i = a j2 > i modulo 4 )

of G. Higman s a tis fie s none of the known su fficien t conditions for the so lvab ility of an equation (with non-zero exponent sum) over G . Without the exponent condition, probing by Lyndon [1 1 3 ] su g g e sts that the problem is difficult even for G a finite c y c lic group.

See a ls o B rodskii [19, 2 0 ]

and Short [152]. The resu lt of G erstenhaber-Rothaus ra is e s the following question. PRO BLEM 2a.

If the sum of the exp o n en ts on t

in w is not 0 , d o e s

the equation w = 1 alw ays have a solution ? So much for equations over groups. fied by the Vaught problem.

Equations in groups are exem pli­

T his prompted Lyndon to a sk about solu tions

of an equation w (a 1 ,- - * ,a n, t^ ,--*, t m) = 1 a l , ’ *‘ , a n*

case

in a free group G with b a sis

m = 1 of one unknown he obtained a s e t of words

containing param eters a s exponents w hich, su b ject to conditions on the p aram eters, give p re cise ly the s e t of so lu tio n s. has e x a ctly the solu tions

t =a^

F o r exam ple,

for all integers

n.

t ^a^ta^ 1 = 1

This was sub­

sta n tially improved and extended, but without definitive resu lt.

R ecen tly

Makanin [115] has given an algorithm that a s s o c ia te s with an equation w = 1 , a s ab ove, an integer N such th at, if any solution e x is t s , there

7

PROBLEMS IN COMBINATORIAL GROUP THEORY

e x is ts one with to ta l length of the t^ a t most N . T his s e ttle s the question of e x is te n c e of solu tion s and provides an algorithm for finding one if it e x is t s , but le a v e s open the following problem. P RO BLEM 3.

G iv en an equation over a fr e e group, find an a lg eb ra ic d escrip tio n o f the s e t o f a ll so lu tio n s.

See Howie [93 , 9 5 , 9 7 ], Ozhigov [1 3 0]. A s p e c ia l c a s e of this problem, for free groups and other groups, is the Substitution Problem (for free groups, the Endomorphism Problem ): D oes an equation w ^ , - - - , ^ ) = g have a solu tion ?

Wicks [1 6 9 ] showed

that an elem ent g of a free group is a commutator if and only if, re lativ e without

to any b a s is ,

g ca n be w ritten in the form g = a b c a ~ 1b“ 1c _1

ca n cella tio n .

T his resu lt has been extended su b stan tially by C .C . Edmunds

[5 1 , 5 2 ] and L . P . Comerford, Jr . [3 4 , 3 5 ] and, using to p o lo g ical methods, by M. C uller [37] and by G oldstein-Turner [7 3 , 74].

R elated methods have

been used by M. Scharlemann [1 4 6 ] and P . E . Schupp [1 4 8 ]; s e e a ls o E . R ips [1 4 0 ], and P . Hill and S. J . Prid e [90]. PRO BLEM 4 .

L e t w (a 1 , - * - , a n) b e a word in the fr e e group F

with

b a s is a p * " > a n * Is th ere an algorithm w hich, g iv en g in F , d e c id e s if th ere e x is t t^ , ***, t n in F —

s u c h that

= g ?

The Substitution Problem has been studied e x ten siv ely for finite sim ple groups and for various c l a s s i c a l infinite groups.

See F in k e lste in

[57], Fin kelstein-M andelberg [58], Lyndon [1 1 2 ], M ycielski [1 2 2 ], and a ls o E h ren feu ch t-Fajtlow icz-M alitz-M ycielsk i [5 3 ].

2.

A utom orphism s of gro u p s The general linear group G L (n r R ) is the group of automorphisms of

the free R-module of rank n . The automorphism group G L (2, Z ) of the free Z module, or free ab elian group, A 2 of rank 2 , and a ls o , in par-

8

ROGER LYNDON

ticu la r, the modular group P S L (2 , Z ) , have been much studied.

But it is

a giant strid e to the study of the automorphism group Aut Fn of the free nonabelian group Fn of rank n . N ielsen used automorphisms of free groups to great ad vantage, and obtained a finite presentation for Aut F . J . McCool recovered this presentation by different methods, which enabled him to obtain finite presen tations for the sta b iliz e rs of finite s e ts of elem ents.

His method

is based on that used by Whitehead to d ecid e whether two finite seq u en ces of elem ents of a free group are equivalent under some automorphism. A related problem of Whitehead, to d ecid e whether two finitely generated subgroups are equivalent under an automorphism, has been solved recen tly by G ersten [68], using new methods.

By the sam e methods he has proved

a co n jectu re of P . S cott that the subgroup of elem ents fixed by an au to ­ morphism of a finitely generated free group is finitely generated.

J . L . Dyer

and S cott had shown earlier that the s e t of fixed points of a finite su b ­ group of Aut F

is a free facto r of F . T here has been some study of the

structure of a sin gle automorphism, inducing the dream, or nightmare, of a ‘Jordan stru ctu re th eo rem .’ C onsiderable work on the structure of Aut F , related mainly to its actio n on various naturally arisin g c h a ra c te ris tic subgroups and their q uotien ts, has been done by S. Andreadakis and by S. Bachm uth-H. Mochizuki. PRO BLEM 5.

D eterm in e the stru ctu re of Aut F , of its su bgro u p s, e s p e c ia lly its fin ite su b gro u p s, and its quotient g ro u p s, a s w ell a s the stru ctu re o f individual autom orphism s.

If N is a normal subgroup of a free group F sta b iliz e r, then A utN F

and A utN F

is its

induces a group of automorphisms of G = F / N .

N ielsen showed th at for the usual presentation of a su rface group G the group A utN F

maps onto Aut G .

G. R osenberger [1 4 3 ] and H. Z ie sch a n g

have studied this Gifting problem ,’ e s p e c ia lly for F u ch sia n groups. a ls o S. J . Pride and A. D. V ella [137].

See

PROBLEMS IN COMBINATORIAL GROUP THEORY

P RO BL E M 6.

If G = F / N ,

F

9

fr e e , what su b gro u p s of Aut G a re

im a ges o f su b gro u p s of Aut F ? The mapping c l a s s group M of a su rfa ce can be identified e s s e n tia lly with the group of outer automorphisms (automorphisms modulo inner au to ­ morphisms) of its fundamental group G . T he resu lts of J . McCool give M (or, d ire ctly ,

Aut G ) a s the fundamental group of a finite com plex,

which, unfortunately, exh ib its too many natural sym m etries to make it a c c e s s ib le to p resent day com putation. W. Thurston (s e e A. H atcherW. Thurston [8 8 ] ) has given a quite different method for obtaining a finite presen tation of the mapping c la s s groups, and, for the orientable case,

B . Wajnryb [1 6 8 ] has used this method to obtain presen tations that

are reasonably c o n c is e but not en tirely persp icu ou s. PRO BLEM 7.

Obtain fin ite p resen ta tio n s for the mapping c la s s groups that a re at o n ce usably c o n c is e and y et in w hich both the gen era to rs and the rela tio n s have fairly obvious g eo m etri­ c a l m ea n in gs.

F o r exam ple, H atcher-T hurston s a y :

“ . .. a ll relation s follow from

relation s supported in ce rta in su b s u rfa ce s, finite in number, of genus at most 2 . ” The following is an obvious addendum. PRO BL E M 7a.

3.

T h e sa m e for a ll F u ch sia n groups.

Morphisms of trees T he prefix ‘a u to ’ is omitted a s a s a lu te to the recen t work of G ersten

[6 2 , 6 4 -7 2 ] and Stallin gs [1 5 8 , 1 5 9 ] on morphisms in the categ o ry of graphs. A b a sic paper of J . T its [1 6 4 ] in itia te s the study of the group Aut T of automorphisms of a tree

T . He show s th at, if Aut T

le a v e s invariant

no proper subtree and no end of T , then the subgroup G generated by

10

ROGER LYNDON

all sta b iliz e rs of branch points is a sim ple group, while the quotient (Aut T )/G groups.

is a free product of groups of order 2 and infinite c y c lic

T his frequently cited work d e se rv e s to be extended.

A great d eal is coming to be known about certain very s p e c ia l but very remarkable groups of automorphisms of tre e s , introduced by N. Gupta and S. Sidki [84, 8 5 , 8 6 , 1 5 3 , 1 5 4 ].

If A is an ‘alp h ab et’ with a prime

number p of le tte rs , then the monoid

T = A*

ordered by left d iv isib ility , is a tree T .

of a ll finite words,

G eneralizing a con stru ction of

R I. Grigorchuk [7 9 ], Gupta and Sidki show that ce rta in e a sily describ ed 2-gen erator subgroups

G of Aut T

have remarkable p roperties:

they are

‘ Burnside g ro u p s,’ that is , infinite 2-gen erator p-groups (with elem ents of unbounded order); they contain isom orphically a ll finite p-groups; they are resid u ally finite, and all their proper quotient groups are fin ite. T h ese groups are remarkable not only for th ese prop erties, but a ls o b ecau se they are quite ‘c o n c re te ’ and a c c e s s ib le to detailed study. PRO BLEM 8.

Study the stru ctu re of the autom orphism groups of trees and of their su b gro u p s.

Fo r exam ple, do th ese groups have Sylow subgroups? C ertain naturally arisin g in sta n ce s of groups a ctin g on tre e s en­ countered by J - P . Serre appear to have led to the B a ss-S e rre theory of graphs of groups, with their a s s o c ia te d groups a ctin g on tre e s.

This

method has becom e a standard tool in the study of infinite groups, e sp e cia lly a s obtained by am algamated product and HN N -extension, and with regard to subgroup theorem s.

E a rlie r, Lyndon, in seek in g to unify

ca n cellatio n arguments based on N ielsen transform ations in the proofs of the N ielsen -S chreier and Kurosh Subgroup Theorem s and of the GrushkoNeumann Theorem , introduced axio m atically ch a ra cteriz e d length func­ tions on groups.

I. M. C hisw ell showed th at this theory, for integer

valued fu n ction s, is e s s e n tia lly equivalent to the B a ss-S e rre theory of groups actin g on tre e s .

J . W. Morgan and P . B . Shalen [1 2 0 ], following

PROBLEMS IN COMBINATORIAL GROUP THEORY

11

work of R .C . Alperin and K .N . Moss [4] on real valued length fu n ction s, have played off th e se two th eories to obtain new proofs of two theorem s of Thurston. In con nection with the Subgroup T heorem s, we note that R osenberger, Z ie sch a n g , and K arrass-S o litar a ll obtained refinem ents of the b a sic Sub­ group Theorem of Hanna Neumann for am algam ated products.

In particular,

K arrass and Solitar introduced tree products, which agree with a s p e c ia l c a s e of the B a ss-S e rre graph products.

They a ls o introduced polygonal

products: a group G is generated by v ertex groups, with the edge groups am algam ated.

(T h is group G differs in a sm all but sig n ifican t way from

the corresponding B a s s-S e rre group; the K arrass-S o litar definition is natural in the co n te x t of p resen tations of ce rta in geom etrically con stru cted groups, while that of B a ss-S e rre is natural for a graph of co m p le x e s.) Polygon al products w ere motivated by the recognition that the P icard group P S L (2 , Z [ i ] ) can be obtained from four very sm all groups at the v e rtice s of a square by am algam ating subgroups a s s o c ia te d with the sid e s of the sq uare.

S ee, for exam ple, B . F in e [5 5 , 5 6 ] and A. Brunner,

M. L . F ram e, Y .W . L e e , N .J . Wielenberg [24].

Square products are

studied a ls o in a paper of D. Z . Djokovic [4 2 ], w hich, although failing of its main o b jectiv e , co n tain s an e x te n siv e study of groups actin g on cu b ic tre e s .

See a ls o Djokovic>G. L . Miller [43].

A. Brunner, Y .W . L e e , and

N .J . W ielenberg [2 5 ] have used polygonal products to obtain elegan t d escrip tio n s of various 3-dim ensional E u clid ean groups.

M. W. D avis [40]

has used a sim ilar con stru ctio n in a more a b s tra c t co n text to obtain gen­ eralized C o xeter groups by sew ing together infinitely many co p ie s of an a b stra ct polytope by id en tification s a t v e rtic e s acco rd in g to sp ecified orthogonal groups; in this way he obtains a sp h e rica l manifolds of dimen­ sion

n> 4

not covered by E u clid ean s p a c e .

T here are a ls o various

a b stra ctly defined gen eralizatio n s of sym m etric groups, braid groups, and C oxeter groups; K. I. Appel and P . E . Schupp [5, 6 ] have used sm all c a n ­ ce lla tio n theory to study ce rta in su ch groups.

12

ROGER LYNDON

PRO BLEM 9.

T h e various co n stru ctio n s m entioned a b o v e appear to o pen the way to a more co m p reh en siv e theory of the s tru c­ ture of in fin ite g ro u p s:

We mention a curious resu lt concerning the question of a c c e s s ib ility by D. E . Muller and P . E . Schupp [1 2 1 ], that a finitely generated group is virtually free if and only if it has a co n text-free word problem and is a c c e s s ib le . See a ls o W. D icks [41], J-C . Hausmann [8 9], A. K arrass-A . P ietrow sk iD. Solitar [1 0 8 ], S tallin gs [1 5 8 ], and M.D. Tretkoff [165].

4.

B u rn sid e groups W. Burnside asked if every finitely generated torsion group is finite.

T his is true for groups of certain sm all exp onen ts, for linear groups, and for analogous L ie rings.

In 1 9 6 0 Golod proved the e x is te n c e of what we

sh all c a ll B u rn sid e g ro u p s, finitely generated infinite torsion groups.

In

1964 P . S. Novikov and S. I. Adian showed that B (m ,p ), the free group on m> 2

generators in the variety of groups of prime exponent p , is infinite

for su fficien tly large

p ; s e e [1].

We have mentioned that Grigorchuk [79]

gave a very sim ple con stru ction for an infinite 3-generator 2-group, and that N .D . Gupta and S. Sidki extended this co n stru ctio n to obtain infinite 2-gen erator p-groups for all primes

p.

B efore Grigorchuk, S. V. A leshin

[3] had con stru cted groups like Grigorchuk’s in terms of autom ata, which Y . I. Merzlyakov [1 1 8 ] showed to be e s se n tia lly equivalent to those of Grigorchuk. A. Y . O lshanskii [1 2 4 , 1 2 5 , 1 2 6 , 1 2 8 , 1 2 9 ] used sm all ca n ce lla tio n theory to co n stru ct infinite 2-gen erator groups, with so lv ab le con ju gacy problem, in which a ll nontrivial proper subgroups are isomorphic c y c lic 7^ groups, a ll infinite c y c lic or a ll of order p , for som e prime p > 1 0 O lsh an sk ii’s groups are T arsk i M onsters, groups a ll of whose proper sub­ groups have sm aller card in ality .

S. Shelah [1 5 1 ], a ls o using sm all c a n c e lla ­

tion theory, had con stru cted e arlier an uncountable T arsk i Monster. E . Rips

PROBLEMS IN COMBINATORIAL GROUP THEORY

13

(s e e [1 4 0 ] ) has obtained re s u lts , not wholly published, p arallel to resu lts of O lshan sk ii, using sim ilar methods. The groups B (m ,p) of Novikov-Adian are u n iversal in the s e n s e of being a s large a s p o ssib le given the number m of generators and the exponent p , and thus they have a sim ple definition, but a t the exp en se of being com p licated to work with. D esp ite this Adian [1] has obtained co n sid erab le d etailed information about them.

The groups of O lshanskii

are more econ o m ical, a s im ages of the Novikov-Adian groups, but their definition is much more com p licated , their p resen tation being d ictated naturally by the problem a t hand; from th is it is e a s ie r to e sta b lish their cru cia l prop erties.

The groups of Gupta-Sidki, although not of finite

exponent, combine the virtu es of being very e a sily defined and relativ ely e a sy to work with. The Burnside problem may be viewed a s a te s t problem, of more in terest in terms of methods than of re su lts .

N o n eth eless, the natural

gen erality of the Novikov-Adian groups together with A d ian ’s d etailed resu lts on them, the fa c t that O lsh an sk ii’s groups of exponent p are infinite finitely generated sim ple groups, and the fa c t that the Gupta-Sidki groups a ris e as groups of automorphisms of tre e s and have in terestin g prop erties, a ll lead to the con clu sio n that B urnside groups are of co n sid e r­ able intrin sic in te re st.

(P erh ap s there was a time when finite p-groups

were con sid ered a s of in terest only a s Sylow g rou p s.) PRO BLEM 10.

D ev elo p a g e n e ra l theory of B u rn sid e gro u p s, in clu d in g a d e ta ile d study of certa in particular s u c h groups.

D esp ite enormous work, mainly for very sm all and very large prime valu es of n , the following problem is far from so lv ed . PRO BLEM 1 0 a .

For w hich pairs m and n is the group B (m ,n) fin ite ?

Another natural problem seem s to be folklore, and little more. PRO BLEM 10b.

Is ev ery fin itely p re s e n t e d torsion group fin ite ?

14

ROGER LYNDON

I have found no referen ce to this problem in print.

Gupta has pointed

out that a ll those who have con stru cted infinite Burnside groups have been a t pains to point out that their groups are not finitely presented. Scott has told me that he and, independently, W. J a c o , had recognized that the e x is te n c e of a finitely presented infinite torsion group would follow from the e x is te n c e of a 3-manifold whose nonabelian fundamental group had an infinitely generated ce n te r. F o r a survey of B urnside groups s e e Gupta [82].

5.

G enerators a n d rela tio n s From the point of view of p resen tatio n s, the sim plest groups after

free groups are the on e-relator groups.

T h e se groups have been studied

e x te n siv e ly , partly b e ca u s e the su rfa ce groups are on e-relator groups, and partly b ecau se they sh are some of the a c c e s s ib ility of free groups while exhibiting co n sid erab le individual com p lexity.

Magnus, giving

cred it to Dehn’s top ological in sigh t, solved the word problem for onerelator groups, and obtained other re su lts by the sam e method, notably the F re ih e its s a tz .

Although his method looks rather e a s y , and perhaps even

obvious from the alg eb raic sid e , in re tro sp e ct, it has yet to be incorporated into a general theory in an entirely s a tis fa c to ry w ay, e sp e cia lly from the to p o log ical point of view .

In con nection with the F re ih e its s a tz , s e e

Lyndon, and B . B au m slag -P rid e, and J . Howie [93].

It is striking th at,

d esp ite the solution of the con ju gacy problem for a fair variety of groups, the con ju gacy problem for on e-relator groups remains unsettled fifty years after M agnus’ solution of the word problem. PRO BLEM 11.

Is the co n ju g a cy problem so lv a b le for a ll one-relator g ro u p s ?

Note that B .B . Newman has solved this problem for a ll one-relator groups with torsion . R ecen tly A. Ju h asz [1 0 3 , 1 0 4 , 1 0 5 , 1 0 6 ] has developed new methods in sm all ca n ce lla tio n theory that so lv e the con ju gacy problem for certain

PROBLEMS IN COMBINATORIAL GROUP THEORY

15

groups, including a new s p e c ia l c la s s of on e-relator groups.

Other recen t

work, e s p e cia lly of S. J . P rid e [1 3 3 , 1 3 4 , 1 3 5 , 1 3 6 ] and of Pride together with P . Hill [90] and with P . Hill and A .D . V ella [9 1 ], has extended sm all ca n ce lla tio n theory to derive co n sid erab le information about the stru ctu re of on e-relator groups and th eir subgroups.

F o r exam ple, Pride

has shown that a on e-relator group with torsion can have only finitely many isomorphism typ es of nonfree tw o-generator subgroups.

See a ls o

G. B aum slag [9] and H .D . Hurwitz [1 0 2 ]. PRO BLEM 1 1 a .

D eterm in e further the stru ctu re of one-relator gro u p s.

Although sm all c a n ce lla tio n theory seem s now to be the b e st tool for studying on e-relator groups and, e s p e c ia lly , their con ju gacy problem, if it turns out th at th ere are on e-relator groups with unsolvable conjugacy problem, it would seem to require very co n sid erab le ingenuity on the sid e of logic to co n stru ct one. A secon d q uestion a risin g sim ply in terms of g en erato rs, initially without referen ce to re la tio n s, is that of the rate of growth on a group. If a group G is generated by a finite s e t X , let elem ents of G represented by words in X

g(n) be the number of

of length a t most n .

J . Milnor,

who introduced th e se id eas in con nection with d ifferential geom etry, asked if the asym p totic rate of growth of g (n ), which does not depend on the c h o ice of X , is alw ays either polynomial (a s for a free ab elian group) or exp onen tial (as for a nonabelian free group), and, with J . A. Wolf, e s ta b ­ lished this co n jectu re for finitely generated so lv ab le groups; for th ese M. Gromov [80] (s e e a ls o A. J . W ilkie-L. van den D ries [1 7 0 ] ) showed th at the growth is polynomial ju st in c a s e the group co n tain s a nilpotent group of finite index.

R ecen tly Grigorchuk has shown that one of his

groups is neither polynomial nor exp onen tial. J.W . Cannon [2 7 ] showed th at for ce rta in F u ch sia n groups and C oxeter groups, with a natural c h o ice of generating s e t

X , the growth generating

function G (z) = 2 g (n )z n is a ration al function, and ch a ra cte riz e d its zero s and p o les.

M. B enson [1 4 ] showed th at G (z) is ratio n al for finite

16

ROGER LYNDON

exten sio n s of free ab elian groups of finite rank. Grigorchuk [78] con sid ered a function

See a ls o M. Grayson [77].

H (z) = X h (n )z n for H a subgroup

of a free group

F , where h(n) is now the number of elem ents of H

minimal length

n re la tiv e

is a rational function.

to a b a s is

F o r H normal and

of

X for F , and showed that G = F /H , he obtains the

sp e ctra l radius

rG of a random walk on G in terms of the radius of

con vergen ce of

H (z ), and proves the following.

integers

M and

T o given

N , there corresponds an integer

e > 0 and

L with the property

th at, if G has a presen tation with M generators and n < N re la to rs, a ll of length a t le a s t condition, then

L , which s a tis fie s a standard sm all ca n ce lla tio n

|rG - \ /2M -l/M | < e, and hence

G is not am enable.

Adian u ses th ese id eas to show that his infinite groups B (m ,p) are not am enable.

C . S eries [1 4 9 , 1 5 0 ] u ses sim ilar methods to study the d is tri­

bution of the s e t of lim its of random walks on F u ch sia n groups. H. B a s s

[8 ], W. J .

PRO BLEM 12.

See a ls o

Floyd [5 9 ], and P . W agreich [167].

T h e re is cle a rly m uch to b e d one in determ ining the p o s s ib le growth fu n ctio n s of groups and in rela tin g them to p ro p erties of groups.

A sim ilarity between the formula for the growth function of a free product and that for the c h a ra c te ris tic su g g e sts a con nection between th ese two co n ce p ts , which is borne out in some of Cannon’s exam p les. It appears th at further understanding is needed of the dependence of the d etailed structure of the growth function on the ch o ice of generating s e t, and, with it, on properties of the s e t of relatio n s. A long standing question is that of the sig n ifica n ce of the d eficien cy of a p resen tation, the e x c e s s of the number of relation s over the number of gen erators.

D eficien cy of a finite group has been much studied.

The

d eficien cy can be viewed a s a truncation of a P o in care s e rie s of a reso lu ­ tion of the p resen tation , and seem s related to the E u le r-P o in ca re c h a ra c ­ te r is tic .

See B . B aum slag-S. J . Pride [10, 1 1 ], K. S. Brown [21, 2 2 ] (and

H (z)

PROBLEMS IN COMBINATORIAL GROUP THEORY

17

the review of [2 1 ] by K.W . Gruenberg [8 1 ] ), I. M. C hisw ell [2 9 ], M. Edjvet [4 9 ], J .G . R a tcliffe [1 3 8 ], N .S . Rom anovskii [1 4 2 ] and R . Stohr [1 6 0 ]. We p ose only a rather vague problem. PRO BLEM 13.

E x te n d a n d re la te the th eo ries of the d e fic ie n c y , the rate of growth, a n d the E u ler-P o in c a re ch a ra c te ristic .

In

particular, what in flu e n c e d o es the d e fic ie n c y have on the stru ctu re of an in fin ite gro u p ? T his seem s an appropriate point to sp eak of ce rta in problems of a d irectly to p o lo g ical origin which have led to eq uivalent or related prob­ lems that can be sta te d in purely group th eo retic term s.

I do not feel it

is my part to do more than mention th e se problems. PRO BLEM 14.

T h e P o in ca re c o n jec tu re .

See G. A. Swarup [1 6 1 ].

Among the many problems in group theory

stim ulated by the P o in ca re co n je ctu re , that of J . J . Andrews and M. L . C urtis is possib ly the b e st known. PRO BLEM 15-

L e t the trivial group have a b a la n ced p resen ta tio n ( X : R ) , w here

jX j = |R| < . C an this p resen ta tio n b e re d u c e d

to a trivial p resen ta tio n by a s u c c e s s io n of transform ations of the follow ing kinds : 1.

N ie ls e n transform ations of X ;

2.

N ie ls e n transform ations of R ;

3.

R ep la cin g an elem en t of R

4.

T ie t z e transform ations introducing (or d e le tin g ) a new elem en t x

of X

by a co n ju ga te;

together with a new relator r in R

d efin in g x ? See R. C raggs [3 6 ], W. M etzler [1 1 9 ], and C . P . Rourke [1 4 5 ]. A geom etrically finite group G is one p o s s e s s in g a finite

K (G ,1 ),

and its geom etric dimension is then the le a s t dimension of su ch a

K (G ,1 ).

18

ROGER LYNDON

G eom etrically finite groups, along with many related co n ce p ts and problems, are d iscu sse d in G. B au m slag -E . Dyer-A. H eller [12]; in co n ­ nection with a related problem of B aum slag-D yer-H eller s e e T . Bortnik [18].

The equality of geom etric dimension and cohom ological dimension

has been proved with one excep tio n . PRO BLEM 16 (T he E ilenberg Problem ).

If G has co h o m o lo gica l dim en-

2, must it a ls o have geo m etric d im ension 2 ? We remark that Baum slag-D yer-H eller exam ine alg e b ra ica lly clo se d groups, using, a s does S .D . Brodskii [1 9 ], a definition that is natural but which differs from the original definition given by W .R . S co tt. A third problem from topology with many group th eoretic con n ection s is W hitehead’s asp h ericity problem. PRO BLEM 17.

Is ev ery su b co m p lex of an a s p h e rica l 2 -com plex a s p h e rica l ?

T his problem has receiv ed con sid erab le attention re ce n tly .

See

R. B row n-J. Huebschmann [23], I. M. C hisw ell-D . J . C ollins-H uebschm ann [30], Collins-H uebschm ann [3 1 ], M. G u tierrez-J. G. R a tcliffe [87], J . Howie [96], Huebschmann [99, 1 0 0 , 1 0 1 ], and A. J . Sieradski [1 5 5 ]. The problems above touch on the large area of q uestions of a homologi­ c a l nature, of which we mention only one more.

T hat is the study of

P o in care duality groups; for th is s e e R. B ieri [2] and, e s p e c ia lly , the talk by B . Eckm ann. We cannot re s is t introducing sm all ca n ce lla tio n theory with a quota­ tion from J . N ielsen (1 9 1 2 ), s e e [123]. “ While in the solution of the generation problem one arranges e le ­ ments in seq u en ce in a ll p ossib le ways — one could sp eak of a linear problem — one rep resen ts the [re la to rs] in the group graph by polygons and in the treatm ent of the identity problem one puts th ese together in a map.

T his latter problem h a s, th erefore, in an exp ressio n

PROBLEMS IN COMBINATORIAL GROUP THEORY

of Dehn’s , ‘ one more dim ension’ than the first.

19

Whether this is a real

or only an apparent d ifferen ce in lev el . .. rem ains u n d ecid ed .” Small ca n ce lla tio n theory has becom e a useful and ubiquitous tool. Applied sm all c a n ce lla tio n theory is exem plified by recen t work of Ju h a s z , O lshanskii, P rid e , and R ip s, who have a ls o , a s w ell a s , for exam ple, Do Long Van [4 4 ], J . Howie-S. J . P rid e [9 8 ], and J . Perraud [1 3 2 ], greatly extended the methods of pure sm all c a n ce lla tio n theory. The theory, a s developed so far, could be d escrib ed a s follow s. From ‘ lo c a l ’ h y p o th eses, on ‘s m a ll’ su bcom plexes of the (2-dim ensional) C ayley com plex (Gruppenbild) of a presen tation (in itia lly , on the s ta rs of f a c e s ) , one d eriv es properties of ‘ la rg e ’ su bcom plexes (singular d is c s , sp h eres, annuli, to ri, . . . ) or of the entire com plex.

T here is an evident

an alogy, or con n ectio n , with corresponding d ifferen tial-in tegral co n c e p ts , for exam ple, in sm all c a n ce lla tio n theory the formulas relatin g ‘ arc le n g th ,’ c u rv a tu re ,’ and ‘ a r e a .’ (Tw o papers in the 4 0 ’s by C. B lan c [1 6 ] and F . F ia la [5 4 ], developing and applying com plex function theory on planar graphs do not seem to have been pursued further.

T here is more

recen t work by P . C artiei on harmonic a n a ly s is on tre e s , and a ls o work on harmonic a n a ly s is on free grou p s.) T h u s, a s w ell as Dehn’s original ap p lication of sm all ca n c e lla tio n theory in the hyperbolic plane to lo g ica l d ecisio n problem s, from which the theory a ro s e , and of the more or le s s obvious co n n ectio n s with topology (for exam ple, a sp h e ricity ) and with purely group th eo retic ‘stru ctu re problem s’ (for exam ple, P rid e ’s work on subgroups), there are co n n ectio n s of an a n a ly tic nature, con cern in g rate of growth, random w alk s, a n a ly sis on groups, e tc . PRO BLEM 18.

E x te n d sm all ca n c e lla tio n theory, e s p e c ia lly in a c c o rd ­ a n c e with the co n n ectio n or analogy with a n a ly s is , and unify the s p e c ia l e x te n s io n s a n d a p p lica tio n s noted a b o v e.

6.

G eom etric groups Much of com binatorial group theory a ro se from the study of F u ch sia n

groups, if one includes su rfa ce groups, and there is no need to mention

20

ROGER LYNDON

the role of hyperbolic groups in the study of higher dim ensional manifolds. In the study of F u ch sia n groups, e sp e c ia lly su rface groups, there seem s to be a long tradition, beginning with Dehn and Magnus, and with N ielsen , and continuing through the work of R eid em eister and Z ie sch a n g and, in an extrem e form, two papers of A .H .M . H oare, A. K a rra ss, and D. Solitar on subgroups of F u ch sia n group, which could be d escrib ed , somewhat invidiously, a s a more or le s s co n scio u s attem pt to free the su b ject from a n a ly sis.

T his tendency is not entirely a matter of p rejudice, in view of

the in creasin g difficulty of applying a n aly tic methods in higher dim ensions and in more a b stra c t situ a tio n s, notably to ‘ g eo m etric’ groups defined com binatorially without any d irect referen ce to an a n a ly tic stru ctu re, as mentioned in S ection 3. To a n on-analytic mind it seem s a great m ystery that com binatorial and an aly tic methods often lead to the sam e re s u lts ; it happens often that o b jects definable com binatorially are realizab le a n a ly tica lly .

A modest

exam ple is the com binatorially regular te s s e lla tio n s of the n-sphere and n -sp ace for n > 2 , and a more im pressive one is the definition by Z ie sch a n g , E . V ogt, and H .D . Coldewey of Richsian groups a s automor­ phism groups of planar 2 -co m p lexes.

So perhaps we should not re je ct

a n a ly s is , but seek to a s sim ila te it. A striking exam ple of such assim ilatio n is J . Cannon’s work based in part on the fa ct th at, if the C ayley com plex for a discontinuous hyper­ b olic group is realized in the natural way in hyperbolic s p a c e , then the co n cep ts of com binatorial (word) and m etric hyperbolic g eo d esic agree about a s well a s could be hoped for.

T his provides a new approach to

sm all ca n ce lla tio n theory a s well a s opening up other v is ta s . Something along the sam e line ap p ears, with a different purpose, in the papers of C . S e rie s.

A geom etrically com binatorial approach appears

in a number of papers by A. L . Edmonds, J . E . Ew ing, and R . S. Kulkarni (e .g . [5 0 ] ) in which they study, for exam ple, torsion free subgroups of finite index in F u ch sia n groups.

Somewhat in the sam e sp irit is the

PROBLEMS IN COMBINATORIAL GROUP THEORY

21

rather ex te n siv e use by a number of workers of c o s e t graphs in the study of Riemann su rfa ce s and of subgroups and quotient groups of F u ch sia n groups and other groups.

S ee, for exam ple, re ce n t work of J . L . Brenner-

R . C . Lyndon and of W.W. Stoth ers. PRO BLEM 19.

D ev elo p a u n ified com binatorial theory of a su itably co m p reh en s iv e c la s s of ‘g e o m e t ric ’ groups.

See a ls o Floyd-H oare-Lyndon [6 0 ] and C . L . and M .D. Tretkoff [1 6 5 ].

7.

A lg eb ra ic rep resen ta tio n s of groups We are now far d ista n t from logic and topology and approaching the

forbidden v a s tn e s s e s of rep resen tation theory, L ie theory, and group rings. B u t, d esp ite the in terdiction by Chandler-M agnus, we must mention Magnus’ rep resen tation of free groups in a s s o c ia tiv e rings, L ie rin gs, and matrix rings.

The first and ce n tra l exam ple here is M agnus’ rep resen ta­

tion of a free group F

in the power se rie s com pletion A of a free

a s s o c ia tiv e ring, carryin g with it a rep resen tation of the descending cen tral quotients of F

by the dim ension modules of the free L ie ring

a s s o c ia te d with A . We refrain from mentioning a ll but one of the many e xten sio n s of th ese id eas. The con nection between the commutator stru ctu re of a group, e sp e cia lly a p-group, and L ie theory, although not a s p erfect a s for co n ­ tinuous groups, has been highly developed, e s p e c ia lly by P . Hall and his follow ers.

C lo sely related to the Magnus rep resen tation is the free differ­

e n tial ca lcu lu s of R . H. F o x , originating in knot theory, but with a homologi­ c a l interpretation along the lines of the work of R eid em eister. Any group G is naturally embedded in its in tegral group ring ZG . L e t A be the fundamental id eal, the kernel of the augm entation map ZG -» Z . The n-th dimension subgroup Dn(G) of G is then defined to be G fl ( 1 + A n) , the group of a ll

g in G such that g = 1 modulo An .

Magnus (1 9 3 5 ) showed that if G is a free group, then Dn(G) = y nG , the

ROGER LYNDON

22

n-th term of the d escending cen tral se rie s of G ; from fl An = 0 it follows that

H y nG = 1 .

It was con jectu red that Dn(Gr) = y nG for a ll groups tive integers

G and a ll p o si­

n , and this co n jectu re is e a sily reduced to the c a s e that G

is a finite p-group.

However, Rips (1 9 7 2 ) exhibited a 2-group G with

y^G = 1 but D4 (G) 4 1 • T his very s p e c ia l counterexam ple, using the only even prime, seem s to leave the door open for a p o ssib le rehabilitation of the problem.

See I. B . S. P a s s i [1 3 1 ], J . A . Sjogren [1 5 6 ], K-I. Tahara

[1 6 2 , 1 6 3 ], and the talk by Gupta. F o x (1 9 5 3 ) introduced the groups a normal subgroup of the free group F natural map Z F -» Z ( F / R ) .

F (n ,R ) = F H ( l + j A n) , where R is and

J

is the kernel of the

The dimension subgroups of G = F / R

represented an alogou sly by the groups

are

D (n,R ) = F D (1 + J + An) . By the

resu lt of Magnus, F ( n ,F ) = Dn+1( F ) = y n+i F > anc* Magnus showed a ls o that F ( l ,R ) = y 2R .

Enright showed that F (2 ,R ) = y 2 (R H y 2F ) y 3R , and

Gupta and P a s s i derived properties of F / F ( n ,R ) from a matrix re p resen ta­ tion.

F o r th ese re su lts s e e the survey a rtic le by Gupta, A problem of

R .H . F o x , C anad. Math. B ull. 2 4 ( 1 9 8 1 ) , 1 2 9 -1 3 6 . PRO BLEM 2 0 .

Obtain in trin sic d escrip tio n s of the F o x su bgroups F (n ,R ) a n d the d im ension su bgro u p s D (n,R ) in terms of the commutator structure of the fr e e group F

and its normal

subgroup R .

A cknow ledgem ent.

The author gratefully acknow ledges partial support of

the N ational S cien ce Foundation. DEPARTMENT OF MATHEMATICS UNIVERSITY OF MICHIGAN ANN ARBOR MICHIGAN 48109-1003

23

PROBLEMS IN COMBINATORIAL GROUP THEORY

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__________. Solution of equations over 1 . Such

a group admits a presen tation G = < x 1 ,y 1 , " - , x g ,y g |[x1 ,y 1 ]---[x g)y g ] = 1 > in the orientable c a s e , G

< Z Q.Zj , " ’ >Zg lZQZl ’ ” Zg = 1 >

in the non-orientable c a s e . We will a lso use the co n cep t of a s u rfa c e group-pair (G; iS Q,S 1 ,--*,S m! ) : it c o n s is ts of the (free) fundamental group of a clo se d su rface of genus g > 0 with m + 1 disks removed, sp h ere), together with the

m > 0 (but > 1

if the su rfa ce is a

m + 1 infinite c y c lic subgroups generated by

the boundary c ir c le s of th e se d isk s.

Surface group-pairs have p resentations

G =t2’-"’tm’Xl'yi’”->Xg'yg> ’ S 0 = < t l t 2 - t m Cx 1 ,y 1 ] - " [ x g ,y g] > , S j = < t j > in the orientable c a s e ,

m+ g > 0 ;

35

for j =

36

BENO ECKMANN

S 0 = < t l t2 * , ' tm z oz l ' " z g > > s j = < t j > for j = in the non-orientable c a s e ,

m > 0,

g > 0.

The “ lo w e st” c a s e s of su rface group-pairs are G = < t 1 >,

SQ= < t 1 >,

S1 = < t 1 >

and G = < z 0 > , S Q = < Z q> . A P o in ca re duality group G of d im ension n , in short a

1. 2.

PD n group,

is a group fulfilling P o in care duality in (c o -) homology for a ll co e fficien t ZG-modules

A with re sp e ct to the formal dimension n and to a certain '■v, G -action on the additive group of integers Z : Hi(G ;A ) 3£ Hn_ j(G ;2 ® A ) ,

Here Z ® A

i fZ .

is a ZG-module by diagonal a ctio n , and the isomorphisms are

natural in the ZG-modules ZG-module structure of Z

A . As we will s e e the dimension n and the n PD -group being

are determined by G (the

orientable or non-orientable accord in g to whether the actio n is triv ial or not). The definition above is , of co u rse , analogous to the P o in care duality valid for a ll clo se d n-dim ensional manifolds being 7Tj(X)-modules.

If X

X , the co e fficie n ts

is asp h e rica l then the (c o -) homology of X

is isomorphic to that of G = 77^ (X ) s o th at, in that c a s e , PD n-group.

A

G is a

It is not known whether in general the co n v erse is true; i .e .,

whether a PD n-group is n e c e s s a rily isomorphic to the fundamental group of a clo sed n-dim ensional a sp h e rica l manifold. Since the u niversal covering of the su rface su rface P D 2 -groups.

is a sp h e rica l.

2 g , g > 1 , is

R 2 the

Thus the su rface groups in 1.1 above are

The T heorem formulated in the title s ta te s that the co n v erse

POINCARE DUALITY GROUPS

is true, thus solving the problem in the c a s e

37

n = 2 . T he proof of that

T heorem has been ach ieved in se v e ra l s te p s contained in a s e rie s of papers by the p resent author and various co lla b o ra to rs; th e se papers were w ritten partly with other o b jectiv es in mind.

The co n feren ce organizers

have asked the author to present a su rvey, a s com plete a s p o ssib le , of that proof. We do so and use the opportunity to sim plify som e of the argum ents. 1 .3 .

There a re , in fa c t, different techniques and methods involved in that

proof.

They belong roughly speaking to the following three a re a s of id e a s:

(I)

H om ological alg eb ra, homology of groups.

(II)

Structure and sp littin g theorem s for groups (S ta llin g s-B a ss-S e rre and o th ers).

(III) Ranks of p rojective modules and E uler c h a ra c te ris tic (H attoriS tallin g s-B a ss-K a p la n s ky). The papers leading to or containing the ste p s of the proof are as follow s.

In the field (I) by Robert B ieri and the author [2 ], [4], [5]; in (II)

by Heinz Muller and the author [1 2 ], [1 4 ]; in (III) by P e te r L in nell and the author [1 0 ], [11]. 1. 4.

F o r com p act manifolds-with-boundary ( (9-manifolds), in particular

for the clo se d su rfa ce s with d isk s removed, one has the well-known “ re la tiv e ” P o in care d uality.

The su rface group-pairs listed in 1.1 above

fulfill su ch a rela tiv e P o in care d uality, of dimension 2 . T o formulate th is in a p re cise way, re la tiv e (co -) homology groups for pairs of groups (G;

,***,SmS) have to be co n sid ered , c f. Section 2 .3 below; this

yields the con cep t of a P D n-pair of groups.

An important step in the proof

of the T heorem will be to show that a ll P D 2 -pairs of groups are su rfa ce group-pairs (“ R ela tiv e T h eo rem ,” S ection 3 .2 ). 1 .5 .

We mention here two co ro lla rie s of the T heorem .

38

BENO ECKMANN

C O R O LLA R Y 1.1 (cf. [1 2 ], [1 1 ]).

A ll P o in c a re -2-co m p lexes a re homotopy

eq u iv alent to c lo s e d s u rfa c e s (of g e n u s > 0 ). C O RO LLA RY 1. 2.

L e t G be a torsion-free group containing a s u rfa c e

group S a s subgroup of fin ite in d ex ; then G is a lso a s u rfa c e group. Indeed, a hom ological argument (s e e Section 2 . 2 ) shows that G is a P D 2 -group.

This corollary is a s p e c ia l c a s e of the “ N ielsen realizatio n

c o n je ctu re ” proved by Kerckhoff [Annals of Math. 1 1 7 (1 9 8 3 ) , 2 3 5 -2 6 5 ]. The s p e cia l c a s e above was estab lish ed by Eckmann-Miiller [12] before K erckhofPs proof, and before our Theorem on P D 2 -groups had been com ­ pletely se ttle d . 1. 6.

T his survey is organized a s follow s.

S ection 2 con tain s general

prelim inaries on duality and relativ e duality groups.

In Section 3 we

assum e that the PD 2 -group fu lfills a certain “ sp littin g ” property and show that the T heorem can then be reduced to the R e la tiv e Theorem ; both that reduction and the proof of the R elativ e Theorem are given on the b a sis of general sp littin g arguments explained in S ection 4.

In Section 5

we show, by quite different methods involving ranks of finitely generated p rojective modules, that any P D 2 -group fulfills the sp littin g assum ption.

2. 2. 1 .

Duality groups The group G is ca lle d a duality group of dim ension n > 0 with

resp ect to a dualizing ZG-module C , in short a Dn group, if one has isomorphisms Hi(G ;A) a for all

i eZ

A , and C ® A

and a ll ZG-modules

Hn i(G; C ® A) A ; they are assum ed to be natural in

is endowed with the diagonal G -action.

Abelian group, one has P o in care duality (cf. 1 .2 ), i .e .,

If C = Z G is a

as an PD n-group.

From the definition it follows that H *(G ;A ) commutes with d irect limits in A . This is p o ssib le only if G admits a projective resolution

POINCARE DUALITY GROUPS

...

^

^ -> ••• -> P Q ->-> Z

with a ll

P-

39

finitely generated over ZG

(by the B ieri-Eckm ann-B row n fin iten ess crite rio n , s e e [3] and [7 ] ). Furtherm ore one e a s ily ch e ck s that C ® Z G = C Q®ZG

(where C Q is the

Abelian group underlying C ) is an induced module, and thus for all

H1(G ;ZG ) = 0

i 4 n ; as for Hn(G ;Z G ), it is isom orphic to HQ(G ;C ® Z G ) = C ,

and this is a (right) ZG-module isomorphism.

The dualizing module C

is thus determined by G ; it is e a s ily se e n to be torsion-free a s an Abelian group. The cohomology dimension cd G is cle a rly it is

< n , and by the above

= n ; hence the integer n is a ls o determined by G , and G admits

a finitely generated p rojective resolution of finite length (equal to

n );

such groups are said to be of type F P . Summarizing we s e e that a Dn-group G fulfills (1) G is of type F P . (2 ) H*(G;ZG) = 0 for i 4 n and Hn(G ;ZG ) is to rsio n -free, (3 ) cd G = n . It has been proved by B ieri-E ckm ann [2] th at, co n v e rse ly , a group fulfilling (1 ) and (2) is a Dn-group with dualizing module C = Hn(G ;Z G ). We note that cd G = n implies that G is torsion -free. 2. 2 .

As an ap p licatio n , let G b e to rsio n -free and S a subgroup of fin ite

index.

By S erre’s theorem (se e e .g . [8 ], p. 1 9 0 ) cd G = cd S = n . C learly

G admits a finitely generated free resolution over ZG does over Z S ; hence

G is of type F P

if and only if S

if and only if S is.

Moreover

Hi(S ;Z S ) ^ Hi(G ;H om s (Z G ,Z S )) s Hi (G ;ZS® ZG) - H *(G ;Z G ), and it follows that G fu lfills (1 ) and (2 ) above if and only if S d o es, with the “ sa m e ” dualizing module C . Thus

G is a Dn-group if and only if S is a Dn-group, and the

dualizing modules are isomorphic a s A belian groups. is a PD n-group if and only if S is.

In p articu lar,

G

BENO ECKMANN

40

1) REMARKS:

D im en sio n 1 .

generated free.

It is a P D 1-group if and only if it is infinite c y c li c .

G is a D 1 -group if and only if it is finitely

2) Subgroups of in fin ite in d ex.

F o r PD 2 -groups G , Strebel [17] has

proved, by hom ological methods, that for a subgroup S of infinite index in G one has cd S < n - 1 . F o r

n =2

it follows that S is a free group;

this is the P D 2 -analogue of a fa ct well known for su rface groups. 2.3.

We briefly re c a ll rela tiv e duality for group pairs (for d e ta ils s e e

B ieri-Eckm ann [4], [ 5 ] ). a family

S

= |Sj, j ^l i

A group pair (G ,S) c o n sists of a group G and

of subgroups, not n e ce ssa rily d istin ct.

subgroup S C G one w rites

Z G /S

for the G-module whose underlying

A belian group is freely generated by the c o s e ts left multiplication.

F o r any

xS , with G -action by

R elative (co-) homology is defined by means of the

“ augmentation k ern el”

A = ker i ©. Z G /S - -^ -> Z ! where

e(xS-) = 1 for

J

a ll x e G and

j eI : Hi(G ,S;A ) = Hi_ 1 (G ;A ® A ) , H ^G .SjA) = Hi - 1 (G; Hom(A,A)) ,

A being a G-module, ® and

Horn equipped with diagonal G -action.

A duality pair of dimension

n with dualizing module C is a pair

(G ,S) fulfilling Hi(G ;A ) ss Hn_ i(G ,S ;C « A ) and Hi(G ,S;A ) a

Hn_ j(G ;C ® A ) .

In the (orien tab le) P o in care duality c a s e ,

C = Z , each of th ese isom or­

phisms implies the other one and the first one becom es H^G jA) S Hn i l (G;A ® A) ; i .e .,

(G ,S) is a P D n-pair of groups if and only if G is a Dtv~1-group

POINCARE DUALITY GROUPS

41

with dualizing module A . R e la tiv e e x a c t se q u e n ce s show that S must be a finite family of P D n - 1 -groups

S Q > S y ,S m .

In p articu lar, if (G ,S) is a PD 2 -pair then G is finitely generated free and

S is a finite family of infinite c y c lic groups.

morphisms yield

The duality is o ­

H2 (G ,S;Z G ) = Z , HX(G ,S;Z G ) = 0 , HX(G ;ZG ) = A . As

shown in [4] the only important hom ological property is

H2 (G ,S;Z G ) = Z ;

indeed, if G is finitely generated free and S a finite family of infinite c y c lic groups it c h a ra c te riz e s PD 2 -p airs.

3. 3. 1.

P D 2 -groups sp littin g over a fin itely g e n e ra te d subgroup A group G is said to sp lit over the subgroup H if either

a non-trivial am algam ated free product G = G- * H N N-extension 1)

G = G,^

H is finitely gen erated ,

(a ) G is

G9 , or (/3) G is an

. Two c a s e s are of s p e c ia l im portance: 2 ) H is finite.

We re ca ll that S ta llin g s’

stru ctu re theorem [1 5 ], [16] for finitely generated groups te lls that 2 ) holds if and only if H1(G ;ZG ) 4 0 . T h e a s s ertio n of the T h eo rem holds if the P D ^-group

PROPOSITION 3. 1.

G sp lits over a fin itely g e n era te d group L . The proof of P rop osition 3 .1 makes strong use of the ‘ ‘sim ultaneous sp littin g theorem ” (in sh o rt: by Heinz Muller [14].

SST) for groups and subgroups, estab lish ed

The SST is a refinement of the relativ e version

(Swan [18], Swarup [1 9 ] ) of S ta llin g s ’ structure theorem ; it d eals with sp littin gs over finite subgroups free only H = 1 will o ccu r.

H , and s in c e in our c a s e

G is torsion -

A short outline of SST and its ap p lication

in the present co n te x t will be given in Section 4 below. The ap p licatio n of SST to P D 2 -groups G = G1

p , with

H1(G ;ZG ) = 0 ,

L

L is

(a ) G = G1 * l G2 , or (/3)

finitely generated, is a s follow s. 4 1 • The index of L

theorem [17] one has cd L < 1 , and hence

Since

in G is infinite; by S treb el’s L

is (finitely gen erated ) free.

We only d escrib e the c a s e (a ), the c a s e (j8) being sim ilar.

42

BENO ECKMANN

If in G = Gj * l G2 the rank of L

is

>1

one h a s, by virtue of SST

(s e e Section 4. 4, A )), sp littin g s Gx = H1 * H2 , L = L j * L 2 with

C H j,

^ , L = h 1 * q L 2 Xq with

G1 = H *< q > = H

i = 1 ,2 ,

L j, L2 C H .

The first p ossib ility yields G = Hx * L1 if L j 4 Hj L 2 . Thus

then G sp lits over

(H2 * G ); L2

L j , and if L j = Hj

then G sp lits over

G sp lits over a subgroup whose rank is le s s than that of L .

The second p o ssib ility yields G = (H *

G )* L1

so

, L 2 ’*

G sp lits over L 2 , again of rank le ss than that of L . Thus we are alw ays reduced to the c a s e where G sp lits over an

infinite c y c lic subgroup C as (a) G = G1 * Since

G2 , or (j8) G =

*

C ,p

C is a P D ^ grou p we can apply the general hom ological argu­

ments of [4], Theorem 8.1 and 8 .3 . group pairs

c

(G ^ C ) and

It follows that in the c a s e (a) the

(G2 ,C ) and in the c a s e Q3 ) the group pair

(G j, !C ,p _ 1 C p !) are PD 2 -p airs.

Now the R ela tiv e T heorem below te lls

that th ese pairs are su rface group-pairs (se e 1. 1) corresponding to clo sed su rfa ce s with one d isk , or two disks re sp e ctiv e ly , removed. In (a ),

G = Gj^

G2 is the fundamental group of the clo sed su rface

obtained by identifying the boundary c ir c le s ;

in (/3),

G = G1 *

C ,p

is the

fundamental group of the clo se d su rface obtained by joining the two boundary c ir c le s by a tube. 3.2.

R e l a t i v e T h e o r e m . A ny P D 2 -pair (G; { S0 , -- -, Smi) is a su rfa c e

group-pair.

43

POINCARE DUALITY GROUPS

r\

Again the proof u se s mentioned in 2. 3.

SST, in addition to the properties of PD -pairs

We proceed by induction on the rank of the finitely

generated free group G . If that rank is H ^ G jZ G ) = Z

1 , i .e .,

s in c e

G =C

G infinite c y c lic

C = ,

is a P D 1-group, and

one has

H1(G ;ZG ) = A sin ce

G is a duality group of dimension 1 with dualizing module A . The e x a c t seq uen ce

then yields S = i < c 2 >f.

A

0

ZG /Sj

©ZG/ S - = Z © Z ; this is p o ssib le only if S = |C,C} or j J Thus the pair (G ,S) is either

(C, {C, Cl)

or

(C=,

1< c 2 >1); i.e . , we obtain p re cise ly the low est orientable c a s e m = 1 , or the low est non-orientable c a s e

g = 0,

g = 0,

m = 0 of the p resen ta­

tion list of su rface group-pairs in 1 .1 . If the rank of G is (a)

> 1 , SST

(se e Section 4. 4 , B )) y ield s sp littin g s

G = Gx * G 2 with S Q = < g 1g 2 > ,

1

4 gj

are (con ju gate to ) subgroups of G1 and

e G j , while the Sj for j > 0

G^ , sa y

S ^ - - - ^ C G1 ,

Sk + l’ " ' = Gx * 1 ; [J with S Q = < p q 1 P^'1g2 > , g l , g2 e G1 , while S l , . . . , S m are (conjugate to) subgroups of G1 . We will deal with (a) only, the c a s e (/3) being sim ilar.

G = (Gj *< g2 >) * , Then S ( ) C G 1 * < g 2 > .

The pair

2

We can write

G2 .

(G2 ; l< g 2 > , Sk+1 ,••• ,Sml) is a P D 2 -pair

by Theorem 8.1 of [4 ]; note that the c a s e argument, and the pair must then be

< g2 > = G2 needs a s p e c ia l

(< g2 > , S < g 2 > , < g 2 >S), cf. [4] p. 5 1 7 .

Similarly the pair ( G1 , {< g^ >, S^^,••• ,S^!) is a PD 2 -pair.

By induction

they are su rface group-pairs, and one e a sily ch eck s that so is (G jiSo,^,-,^)-

44

BENO ECKMANN

Sim ultaneous sp littin g of groups and su bgroups

4. 4. 1.

We con sid er throughout this se ctio n a finitely generated group G

which sp lits over a finite subgroup K a s &

*

i .e ., with H1(G ;ZG )

4 0,

F

cf. Section 3. 1.

G = G. * 1

K

G0 2

or G = G. *

1 K,p

,

Given a finite family

of subgroups of G let N( G; S1 , -*-,Sm) be the in tersectio n of the kernels of the re strictio n maps resj : H ^ G jZ G ) -> H1 (S j;Z G ), j = l , - - - , m.

By the relativ e version of the structure theorem (Swan [18],

Swarup [1 9 ] ) N (G ;S1 ,--*,Sm)

40

if and only if there is a sp littin g of G

su ch that the Sj are conjugate to subgroups of G1 or G2 ; this will be the c a s e in the following and only such sp littin g s will be con sid ered . L e t T be a further subgroup of G , and assu m e that T generated.

is finitely

Sx^i denoting a s e t of c o s e t rep resen tativ es of G mod T , we

con sid er the restrictio n map

H ^ G jZ G ) -!§§» H1 (T ;Z G ) s

H 1( T ; Z T ) x i / .

The minimal number of non-zero components of r e s (c ) for a ll 0 4 c e N(G;S1 ,***,Sm) is ca lle d the w eight n(T) of T with re sp e ct to G (and to S 1 ,***,Sm) .

Note that n (T ) = 0 if and only if N (G ;T ,S 1 ,• **,Sm)

4 0;

i. e. , if there is a sp littin g of G with T C G^ or G2 . 4.2.

The sim ultaneous sp littin g theorem (SST) estab lish ed by H. Muller

[14] (actu ally for more general G and T ) con cern s the c a s e

n (T ) > 0 .

It can be formulated roughly a s follow s, its full content being in fa ct more com p licated . There is a tree T

on which

G a c ts with finite e d g e -sta b iliz e rs and

with proper subgroups of G a s v e rte x -sta b iliz e rs such that T /G one edge (and that e a ch r

Sj , j = l , - - - , m s ta b iliz e s a v ertex of T ); and

con tain s a su btree T t

n (T) ed ges.

has

invariant under T with T t / T

having at most

45

POINCARE DUALITY GROUPS

EXA M PLES.

1)

Let

G be to rsio n -free, and

n (T ) =

1.

Then one has the

following p o s s ib ilitie s :

(1) G = Gx *G2 ,

T = Tx*T2 , T j C G j , T2 CGr

(2)

= G1 * i, T = T 1 * p T 2 p_ 1 , T j . T j C G j .

G = G1 * < p >

(3) G = < p > , 2)

T=

,

L e t G be to rsio n -free,

S1 = - - - = S m = l T

or m = 0 .

infinite c y c li c , and

n (T ) = 2 . Then one

has the following p o ssib ilitie s (cf. [1 4 ] ) (1) 0 = 0 ^ 2 , (2)

G = Gj * < p >

(3) G = < p > , 4 .3 .

T = < g l g2 > ,

1/gjfG j,

i = 1, 2 •

= Gx T = < p g 1p ' 1g 2 > ,

T = < p 2 >,

S 1 = " - = Sm = l

gr g2

fG j .

or m = 0 .

We re s tric t ou rselv es to some remarks concerning the proof of SST.

We w rite Z 2 for Z / 2 Z

and u se

Z 2G , Z 2 T

instead of Z G , Z T ; one

e a sily ch e ck s that this yields the sam e weight n ( T ) . Then H1(G ;Z 2 G) can be interpreted a s group of a ll su b se ts of G which are “ alm ost in­ v a ria n t” under tran slatio n without being G or 0 ; alm ost invariant means invariant e x ce p t for finite s e ts . 11 Z 0x and x H ° ( G ; eG) - H ^ G jZ G )

H ^ G jZ ^ G ) = 0

yield s ( eG ) g / ( Z 2 G ) g a= H1 (G ;Z 2 G) . The restrictio n map H1 (G ;Z 2 G) U H> U H T x ^

for a ll v where

e ^ H 1( T ;Z T ) x j/ is then given by U is a non-trivial alm ost invariant s e t.

46

BENO ECKMANN

The com ponents

U fl T x^ are alm ost T -in variant.

By the techniques of

Dunwoody [9] and Swarup [19] th ese alm ost invariant s e ts yield the theorem. 4. 4.

There remains to compute the w eights of th ose subgroups which

occur in the proofs of P roposition 3.1 (subgroup L ) and of the “ R e la tiv e Theorem ” in 3 .2 (subgroup S Q ). A) We again re s tric t ou rselves to the c a s e (a) in the proof of Proposition 3. 1.

Thus

G is a P D 2-group with G = G1 * G9 , L free of rank > 1 . * L ^

We claim that

n (L ) with re sp e ct to

(G ^ 0 )

or to (G2 ; 0 )

is equal to 1 .

By example 1) in 4 .2 this yields the required sim ultaneous sp littin g s of G1 and

L.

T o prove the claim we con sid er the e x a c t M ayer-Vietoris seq u en ce i i (r e s 1- r e s 0 ) 1 § 0 ••• -> 0 - H1(G1 ;Z G )© H 1(G2 ;Z G ) i HX(L ;Z G ) ~^>H2 (G ;ZG ) and note that the weight is not 0 sin ce H1(L ;Z L )

res^

and

is free A belian of infinite rank sin c e

Thus the re strictio n of 8

L

r e s 2 are in je ctiv e . is free of rank > 1 .

to H1 ( L ;Z L ) cannot be in je ctiv e ,

being = Z ; i .e ., the in tersectio n

H2 (G ;ZG )

im (r e s 1 , - r e s 2 ) fl H ^ L jZ L )

On the other hand, if both n (L ) with re sp e ct to

•••

is

^0.

(G ^ ,0 ) and to (G2 , 0 )

are > 1 , the image r e s 1( c 1 ) - r e s 2 ( c 2 ) of H1(G2 ;Z G ) cannot lie in H1( L ;Z L ) C H ^ L jZ G ); this is se e n by looking a t the lengths of elem ents with re sp e ct to c o s e t rep resen tativ es of G mod G1 and

G2 . Thus

n(T ) with re sp e ct to , s a y ,

(G ^ 0 )

is

=1 ,

and we obtain the sim ultaneous sp littin g . B ) In 3 . 2,

(G ,S) is a P D 2 -pair,

S = i SQ, S1 ,*• *,Sm! with m > 0

and a ll

Sj infinite c y c li c , and we have to con sid er the c a s e where the free group G is of rank > 1 . The claim is that n(SQ) with re sp e ct to (G, J is

= 2 . By exam ple 2 ) in 4 .3 this yields the required sp littin g s.

, — ,Sm J)

POINCARE DUALITY GROUPS

47

The e x a c t rela tiv e cohomology seq u en ce of G mod S is m

0 — » (G ,S;Z G ) — » H1 (G ;ZG )

cs

® H ^ S -jZ G ) j=0

where r d enotes the map with components P D 2 -pair properties te ll that the first term is

H2 (G ,S;Z G ) —* 0 ,

r e s j , j = 0, l , - - - , ni .

0 , the la s t isomorphic to

Z . If S Q (or any proper su b se t) is omitted from the family term becom es

non-zero.

S , the la st

40

(cf. [4], S ection 1 1 );

N = N (G ;S1 ,*-*,Sm) of the

ker resj , j = l , - - - , m

0 and the first term must be

i . e. , the in tersectio n

The

is

The weight n(SQ) is the minimal number of components in

H ^ S q jZ G ) = ®^H1 (S 0 ;Z S 0)x^/ = © ^Zx^ of r e s Q(c ) for a ll

O^ceN;

note that ker r e s Q fl N = 0 . Now r(N) = ( r e s o(N ) ,0 ,- '- ,0 ) = (H1 (S o;Z G ) ,0 ,- " ,0 ) 0 ker S . e a sily ch e ck s th at S re stricte d to any summand Z x^ b ije ctiv e .

One

of H ^ S q jZ G ) is

So obviously the minimum number of components of elem ents

40

in r e s Q(N) is

5.

T h e first B etti number of a P D ^-group

5. 1.

In order to con clu d e the proof of the Main T heorem we have to show

2 , which proves the claim .

that a P D 2 -group, without any further assum ption, sp lits over a finitely generated subgroup.

T his is guaranteed by the following proposition

w hose proof w ill be given in 5 .2 and 5 .3 below. are of type F P

R e c a ll that PD n-groups

so that B e tti numbers j3j(G) = rank H^(G;Z) are defined

P r o p o s i t i o n 5. 1.

If G is a P D 2 -group then j81( G ) > 0 .

From this it follow s that H1( G; Z) , the ab elian ized group G , co n ­ tain s at le a s t one infinite c y c lic summand C , and thus

G admits a

facto r group = C . By a resu lt of B ieri-S treb el ([6 ], Theorem A) this im­ p lies that G sp lits a s L

G = G^

over a finitely generated subgroup

(as shown in [6], th is holds for any group G of type F P 2 , i .e ., ad ­

mitting a p ro jectiv e resolution which is finitely generated in dim ensions

48

BENO ECKMANN

< 2 , and having an infinite c y c lic factor group).

The splittin g is co n ­

structed e x p licitly , the generator p being any elem ent projecting onto a generator of C . 5 . 2.

For the proof of Proposition 5.1 we use the hom ological Euler

c h a ra c te ris tic

x (G ) =

0(G )-

2 (G) of G .

By the E u ler-P o in care

formula ^ (G ) is equal to the altern atin g sum of the ranks of the free Abelian groups Z ®G P .

for any F P -re so lu tio n over ZG

We re ca ll that in the orientable c a s e even; in the non-orientable c a s e

/3 Q(G) = 1 , /32 (G) = 0 (the proofs are

the sam e a s for clo se d s u rfa c e s). Proposition 5.1 is

y (G ) < 0 .

/3 Q(G) = /32 (G) = 1 and /3X(G) =

Thus in both c a s e s the claim of

If G is a non-orientable PD 2 -group, let

G q be the orientable subgroup of index 2 . By the m ultiplicative property of the E u ler c h a ra c te ris tic (valid for groups of type F P , c f. [8], § IX .6 ) X (G 0) = 2 x ( G ) ; hence x ( G0) < 0 implies

y (G ) < 0 , and we are reduced

to the orientable c a s e . G being an orientable P D 2 -group we ch o o se a resolution

with

ZGd

0 — >P

(1)

ZG

z

P finitely generated projective over ZG . Applying HomG(-,Z G )

to (1 ) we get the seq uen ce

(2 )

ZG 0. BENO ECKM ANN M A T H E M A T IK E T H -Z E N T R U M C H -8 0 9 2 Z U R IC H S W IT Z E R L A N D

REFERENCES [1]

B a s s , H., Euler c h a ra c te r is tic and c h arac ters of d iscre te groups. Inventiones Math. 3 5 ( 1 9 7 6 ) , 1 5 5 -1 9 6 .

[2]

B ie ri, R ., and Eckmann, B . , Groups with homological cuality gen­ eralizing P oin care duality. Inventiones Math. 2 0 ( 1 9 7 3 ) , 1 03 -1 2 4 .

[3]

______ ___ , F in ite n e s s properties of duality groups. Helv. 4 9 ( 1 9 7 4 ) , 4 6 0 -4 7 8 .

[4]

, R elativ e homology and Poincare duality for group pairs. J . of Pure and Applied Algebra 1 3 ( 1 9 7 8 ) , 2 7 7 -3 1 9 .

[5]

, Two-dimensional Poincare duality groups and pairs, in: Homological Group Theory, London Math. Soc. Lecture Notes 3 6 ( 1 9 7 9 ) , 2 2 5 -2 3 0 .

[6]

B ieri, R ., and Strebel, R ., Almost finitely presented soluble groups. Comment. Math. Helv. 5 3 ( 1 9 7 8 ) , 2 5 8 -2 7 8 .

[7]

Brown, K. S., Homological criteria for finiten ess. Helv. 5 0 ( 1 9 7 5 ) , 12 9 -1 3 5 .

[8] [9]

, Cohomology of Groups.

Comment. Math.

Comment. Math.

Springer Verlag New York 1 9 82 .

Dunwoody, M. J . , A c c e s s ib ility and groups of cohomological dimen­ sion one. P roc. of the London Math. Soc. 3 8 ( 1 9 7 9 ) , 1 9 3 -2 1 5 .

[10] Eckmann, B . , and Linnell, P . , Groupes a dualite de P oincare de dimension 2. C .R . Acad. Sci. P aris 2 9 5 ( 1 9 8 2 ) , Serie I, 4 1 7 -4 1 8 .

POINCARE DUALITY GROUPS

51

[11] Eckmann, B . , and Lin nell, P . , P o in care duality groups of dimension two, II. Comment. Math. Helv. 5 8 ( 1 9 8 3 ) , 1 1 1 -1 1 4 . [12] Eckmann, B . , and Muller, H., P o in ca re duality groups of dimension two. Comment. Math. Helv. 5 5 ( 1 9 8 0 ) , 5 1 0 -5 2 0 . [13] Montgomery, S., L eft and right inverses in group alge b ras. Amer. Math. Soc. 7 5 ( 1 9 6 9 ) , 5 3 9 -5 4 0 . [14] Muller, H., Decom position theorems for group pairs. 1 7 6 ( 1 9 8 1 ) , 2 2 3 -2 4 6 .

Bull.

Math. Z eitsch rift

[15] Stallings, J . R ., On torsion-free groups with infinitely many ends. Ann. of Math. 8 8 ( 1 9 6 8 ) , 3 1 2 -3 3 4 . [16]

__________ , Group theory and three-dimensional manifolds. Monographs 4 , Y a le Univ. P r e s s 1 9 7 1 .

Y a le Math.

[17] Strebel, R ., A remark on subgroups of infinite index in Poincare duality groups. Comment. Math. Helv. 5 2 ( 1 9 7 7 ) , 3 1 7 -3 2 4 . [18] Swan, R. G., Groups of cohomological dimension one. 1 2 ( 1 9 6 9 ) , 5 8 5 -6 0 1 .

J . Algebra

[19] Swarup, G. A ., R e la tiv e version of a theorem of Stallings. and Applied Algebra 11 (1 97 7 ), 7 5 -8 2.

J . of Pure

HOW TO G E N E R A L IZ E O N E-R E L A T O R GROUP THEORY Ja m es Howie1 The primary purpose of this paper is to survey a number of recent results in combinatorial group theory. in the title:

The main theme is the one indicated

that of trying to g eneralize the rich theory of one-relator

groups to a more general construction known a s the one-relator (or anom alous) product.

There are a l s o a number of subplots.

F o r example,

I want to illustrate the use of a geometric technique from 3-manifold topology—that of (P apakyriakopoulos)—towers in combinatorial group theory.

The rele va n ce of one-relator theory here is that the c l a s s i c a l

one-relator technique initiated by Magnus [38] is just the tower technique in d isg u ise . v a rie tie s.

I a ls o want to d is c u s s equations over groups and q uasi­

L a s t , but not le a s t, I want to give some publicity to the c l a s s

of locally indicable groups, which I believe to be of fundamental impor­ ta n ce in low-dimensional topology. Our story begins in 1 9 8 0 , with a 1-page announcement [6] by Sergei Brodskii of the following three strong resu lts. THEOREM 1.

T o rsio n -free 1 -relator groups a re lo ca lly in d ica b le.

THEOREM 2.

A ny n o n -d eg en era te equation over a lo cally in d ica b le group

has a solution in so m e overgroup. THEOREM 3.

T h e re e x is t s a nontrivial q u a siv a riety

under e x t e n s io n s , s u c h that ev ery group in 3 eq uationally c l o s e d 2 group in

S’

of g ro u p s, c lo s e d

ca n be em b ed d ed in an

S’ .

1 S u p p o rte d b y a n S E R C A d v a n c e d F e llo w s h ip .

2

T h e R u s s ia n te rm is * ‘ a lg e b ra ic h e s k i z a m k n u ty ,” w h ic h m e ans l i t e r a l l y “ a lg e ­ b r a i c a ll y c l o s e d , ” a n d is u s u a lly t r a n s la te d as s u c h (s e e [6 ] a n d [3 2 ], p. 2 , Q u e s tio n 2 ). U n fo r tu n a te ly th e re a lre a d y e x is t s a d i s t i n c t n o tio n o f “ a lg e b r a ic a lly c lo s e d g r o u p ” [5 7 , 4 3 ] in E n g lis h . I am t r y in g to a v o id c o n fu s io n b e tw e e n th e s e tw o n o tio n s .

53

54

JAMES HOWIE

By the time [6] appeared, I had independently discovered Theorems 1 and 2 [21, 2 2 ], and Hamish Short had independently discovered Theorem 2 [58].

B ro d sk ii’s proofs were published only recently [5].

One interesting

factor is that his arguments are purely a lg eb raic, while Short and I use (different) geometric methods.

B. Baumslag [2] later rediscovered

B ro d skii’s proof of Theorem 2, which e ssen tia lly follows Magnus’ proof of the F r e i h e i ts s a tz [38]. The paper is organized a s follows. briefly d i s c u s s , resp ectively :

In se ctio n s 1 - 3 below I shall

locally indicable groups; one-relator

products and equations over groups; and q uasivarieties.

In se ction 4 I

sh all introduce the notion of a tower of 2-co m plexes, and sketch a proof of the c l a s s i c a l F r e i h e i ts s a tz to illustrate their use.

In sectio n 5 I shall

list a number of resu lts, and in sectio n 6 I shall d iscu ss some open problems. Almost all the results mentioned in this paper have appeared (or will soon appear) elsew here, so I will for the most part omit proofs.

Lemmas

2 and 3 in sec tio n 4 and Theorem 16 in sec tio n 5 are new, and I will indicate how to prove them. This is an expanded version of my lecture at the conference.

I have

deliberately included more material here than I could present in a single lecture, in the hope of broadening the scop e of the a rticle .

Much of this

material was contained (in a very unpolished form, and complete with gory details of proofs) in a co u rse of lectures given at the University of Glasgow in 1982.

I am grateful to the audience there for their fortitude.

In particular I am grateful to Steve Pride for reading and commenting on a draft version of the paper.

I am a ls o grateful to Steve Gersten for further

helpful comments.

1.

What a re loca lly in d ica b le gro u p s, and why a re they in te re s tin g ? A group G is in d ica b le (or X -in d ica b le) if H1 (G ,Z) = Hom (G ,Z) ^ 0 .

In other words, if there e x is t s an epimorphism G -» Z (called an in d exin g

ONE-RELATOR GROUP THEORY

fu n ctio n ).

55

A group is lo ca lly in d ica b le if every nontrivial, finitely gen­

erated subgroup is indicable. group is torsion free.

Thus, for example, every locally indicable

T h e se groups first appeared in Higman’s th esis [20]

(se e a ls o [19, 5 0 ] ) on group rings. THEOREM 4 [19]. in d ica b le group.

L et R

be an in tegra l domain and G a locally

T h e n the

idem potents other than 0

group a lgeb ra RG

has no z ero d iv iso rs, no

and 1 , a nd no units other than th o se of the

form ug ( u a unit in R , g e G ). Higman’s results have subsequently been extended to larger c l a s s e s of groups.

(See for example [47] for d e ta il s .)

Theorem 4 holds whenever

One example is that

G is right o rderable.

In other words

G

admits a total order relation < which is invariant under right multiplica­ tion x < y '=5> x z < y z . THEO REM

5 [7].

L o c a lly in d ica b le groups a re right o rderable.

Here are two further ch aracte riz ation s of locally indicable groups. his paper [ l ] on Whitehead’s Conjecture [67] that any subcomplex an a sp herical 2-complex

L

is a sp h e rica l,

of co n serv a tiv e (or Z -co n serv a tiv e) groups.

co n cept in a cohomological form: ® (Z )

K of

Adams introduced the c l a s s A group G is co n serv a tiv e

if, whenever G a c t s freely and cellularly on a 2-complex H2 ( X / G , Z ) = 0 , then H2 ( X , Z ) = 0 .

In

X

such that

Strebel [61, 6 2] rediscovered this

say that G is a ® group, or G e ® =

if, whenever f : P -» Q is a ZG-homomorphism between projective

ZG-modules, with Ker (1 ® f ) = 0 , then Ker f = 0 , where l ® f : Z ® G P-> z ®g Q • THEOREM 6 [13, 29 ].

G is lo ca lly in d ica b le

G is co n serv a tiv e

G G are in jective.

Fo r more

general one-relator products, we will say that the F re ih e its s a tz holds (in a given situation) if A -> G , B -> G a re injective.

Unfortunately, the

F r e i h e i ts s a tz does not alw ays hold. A , B be simple groups, and let a e A , b e B be e l e ­

Let

EXAM PLE.

ments of d istinct finite orders.

Then

(A * B ) / N ( a b ) = i l i .

This example clearly represents the worst possible pathology, from the point of view of one-relator products.

Since the F r e i h e i ts s a tz is

fundamental to one-relator group theory, we cannot hope to pursue our program of generalization without imposing some conditions to avoid su ch examples.

One possibility is a condition on the relator.

THEOREM

8 [16].

T h e F re ih e its sa tz holds w h en ev er r = s m for som e

m> 4 . We sh all be concentrating, however, on the other possibility, namely imposing re strictions on the fa c to rs.

The following “ folklore” conjecture

is one plausible example. C O N JE C TU R E .

T h e F re ih eitss a tz holds for torsion fr e e fa cto rs.

At present this is merely a con jectu re.

None of the known methods of

a tt a c k apply to torsion free groups in full generality. res u lts, a stronger restriction is needed.

To actu ally obtain

There are generalizations of

the F r e i h e i ts s a tz in [3, 15 , 3 5 , 4 8 ], but the following is the most general form to date.

58

JAMES HOWIE

THEOREM 2 ' [6, 2 1 , 58].

T h e F reih e its sa tz holds for lo ca lly in d ica b le

factors. It turns out that the condition of local indicability is just the right one to make proofs work—not just of the F r e i h e i ts s a tz , but a lso , a s we will s e e later, of other results from one-relator group theory.

T h e se re ­

s u lts , in fa ct, can all be extended to constructions more general than onerelator products, namely to the fundamental groups of s ta g g e re d g e n e ra liz e d 2 -co m p lex es in the se n s e of [28].

I shall not go into d etails here, but

these groups all have the form ( *• A-)/NSrj \, where the Aassumed locally indicable, and the relators

are to be

rj are “ s ta g g e r e d .”

This is

a natural generalization of staggered presentations, and a ls o of s ta g g e re d 2 -c o m p le x e s , which will be defined in s e c tio n 4 below.

An interesting

sp e cia l c a s e is the tree anom alous product of Brodskii [5]. factors

A-

Here the

correspond to the vertices of a tree whose edges correspond

to the relators

rj , and each

rj

involves only the two factors correspond­

ing to the endpoints of the corresponding edge. As the numeration s u g g e s ts, Theorems 2 and 2 ' are c lo s e ly related. To explain this relationship, I must now d is c u s s equations over groups. Consider a one-relator product G = (A * B ) / N ( r ) infinite c y c l i c .

We can regard

in which B = < t >

r as a “ polynomial”

is

r(t) in the variable

t , with co efficien ts from the group A . The polynomial equation r(t) = 1 has a solution in A if and only if the natural map A -> G is split in ­ je c t iv e .

If this map is merely in je ctiv e , then we can regard

overgroup of A in which the equation has a solution. solution is universa l in the obvious s e n s e :

to h e H . (H ence, in particular,

A

Moreover, this

if H is any overgroup of A

in which the equation r(t) = 1 has a solution A -> H factors uniquely through A

G a s an

h , then the inclusion

G in such a way that t e G is sent G is in je ctiv e.) We can think of

G as being formed from A by “ adjoining a root” of the polynomial r ( t ) . T h ese ideas are due originally to B. H. Neumann [42].

59

ONE RELATOR GROUP THEORY

C all a polynomial r = r(t) n o n d eg en era te if r does not belong to any conjugate of A , and c a ll an equation nondegenerate.

r(t) = 1

n o n d eg en era te if r is

Now Theorem 2 ' clearly implies Theorem 2 (put B = < t > ).

C onversely, Theorem 2 a ls o implies Theorem 2 ' : the “ tw iste d ” embedding A * B = A * ( t B t _ 1 ) c — > (A * B ) * < t > of length a t lea st 2

in (A * B )

sends any c y c li c a lly reduced word

to a nondegenerate polynomial.

We can a ls o co nsider the idea of a system

ri(ti» -.tn) = “ •=rm(tl> - ’tn) = 1 of se v e ra l equations in s e v e ra l unknowns, over a group A . It is then appropriate to study the group G = (A * < t 1 , •••,tn > )/N lr 1 ,-- -,r mS, and the natural map A -> G . A group A is called a lg eb ra ica lly c lo s e d [43, 57] (or ex isten tia lly c l o s e d ) if A ^ 1 and any finite system

2

of equations

over A , having a solution in some overgroup, has a solution in A . (In other words, whenever the map A

G is injective in the above, then it

is split in je c tiv e .) A group A is called equationally c lo s e d if every (single) nondegenerate equation over A has a solution in A . (In other words the map A -> (A * < t > ) / N ( r )

is always split in jec tiv e.) It is not too

difficult to s e e that equationally c lo s e d groups are torsion free, while a lg eb ra ica lly c lo s e d groups contain elements of all finite orders, s o that no group falls into both c l a s s e s (cf. footnote 2). T o any fin ite system

X of equations over a group, we can naturally

a s s o c i a t e an integer matrix

M(£) = (/zy), where /z-

is the algeb raic sum

of the exponents of t- appearing in the word rj . If rank M(2) is equal to the number of equations in 2 , then we c a l l or nonsingular. £

2

( lin ea rly ) in d ep en d en t

If this remains true working modulo some prime p , then

is called p-in d ep en d en t.

THEO REM

9 [14].

E v ery in d ep en d en t sy stem of equ a tio n s over a com pact,

c o n n e c te d L ie group A has a solution in A .

60

JAMES HOWIE

CO R O LLAR Y

1 [14].

E v ery in d ep en d en t sy stem of equations over a

fin ite group has a solution in som e fin ite over group. C O R O LLAR Y

2 [49].

E v e ry in d ep en d en t sy stem of equations over a

locally resid u a lly fin ite group has a solution in som e overgroup. THEOREM 10 [13].

E v e ry p -independent sy stem of eq u a tio n s over a

lo ca lly p-ind ica b le group has a solution in som e overgroup. C O RO LLA RY [21].

E v ery in d ep en d en t sy stem of equations over a locally

in d ica b le group has a solution in som e overgroup. There are a ls o numerous results about equations over groups based on restrictions on the equations [11, 2 3 , 33, 4 2 , 52, 5 3, 54] (se e a ls o [37], Chapter 1.6, and the survey a rtic le [36] of Lyndon).

A number of other

results and conjectu res related to th ese problems are d isc u ss ed in [25, 60].

3.

Q u a siv a rieties Varieties of groups have been extensively studied (s e e for example

[ 4 4 ] ).

A variety is a c l a s s of groups defined by some co lle ctio n of laws

or id en tities, for example the variety of metabelian groups is defined by the single identity [[a,b], [c,d ]] = 1 . Similarly a quasivariety is a c l a s s of groups defined by a c o lle ctio n of q u a si-id en tities (in which implication signs may appear).

Thus for example the c l a s s

3?

of torsion-free groups

is a quasivariety defined by the colle ctio n gn = 1

g = 1 (n = 2 , 3 , 4 , - " )

of quasi-identities. All the above can be made p recise using the language of predicate ca lc u lu s.

F o r d etails s e e [39], Chapter V.

Alternatively, a c l a s s

A of

groups is a quasivariety if and only if it contains the trivial group and is clo sed under subgroups, direct products and ultra products. nontrivial notion here is that of an ultraproduct.

The only

The reader is referred to

[39], Chapters IV-V for a comprehensive introduction, but here is a brief sketch.

ONE-RELATOR GROUP THEORY

L e t I be a s e t .

61

An ultrafilter U on I is a subset

of the power s e t

of I which s a t i s f i e s : (i)

X , Y £ U =^> X fl Y 6 U ;

(ii)

X £U, XCY => Y £U ;

(iii)

X C I ==> p r e c is e ly one ofX ,

I-X

belongs

Obvious examples are the p rincipal ultrafilters

to U . U

= {X C I ; x £ Xi

for x £ I , but th e se are not interesting from our point of view.

If I is

finite, then all ultrafilters on I are principal, but otherwise examples of nonprincipal ultrafilters c a n be found using Z o rn ’s Lemma. Now let

{G^S be a family of groups indexed by I .

The ultraproduct

of the G- defined by U is the group G = (11^ G ^ ) /~ , where ~

is the

equivalence relation (gj) ~ (hi) Fo r example

and only if (i £ l : gi = hi ) £ U .

G = Gx whenever

U is the principal ultrafilter

Ux .

(This is why principal ultrafilters are uninteresting.) It is an e a s y e x e r ­ c i s e to c h e ck that the c l a s s e s

2) , ®p , 3 1 0 , J ®

of locally indicable,

locally p-indicable, right orderable and torsion-free groups, re sp ectiv ely , are close d under the formation of ultraproducts (and hence q uasiv arieties). It is not too difficult to write down exp licit s e t s of quasi-identities d e ­ fining $ , ®p and (as we have se e n ) J ® , but I do not know of an exp licit s e t of quasi-identities defining 5 1 0 . Ultraproducts c a n be very powerful.

An interesting application is the

proof of van den Dries and Wilkie [65] of Gromov’s Theorem on groups of polynomial growth. A class

3

of groups is idem potent or e x te n s io n -c lo s e d if 3 * 3 = 3 , in

other words if any extension of an 3-group by an 3-group is an 3-group. Brodskii [5] co n stru cts a quasivariety (i)

$

of groups satisfyin g:

Z fS ;

(ii) (iii) 2

is minimal with r e sp e c t to (i) and (ii).

62

JAMES HOWIE

(Indeed properties ( i ) - ( i i i ) uniquely determine d .) He then proves the following result. THEOREM

3 [5, 6].

T h e q u a siv a riety 3

e x te n sio n s, and ev ery group in 3

is nontrivial and c lo s e d under

ca n he em b ed d ed in an equationally

c lo s e d group in 3 . This answ ers, in an extremely strong way, a question of Bokut' [32], Question 2, p. 2:

does there e xist a nontrivial, equationally clo sed group?

It would be more sa tisfa cto ry to have an exp licit description of the quasivariety

3 . There is a chain of inclusions & c $ c R 0 c .T5F

of which the first follows from property (iii) of 3 Theorem 5. 3 .3 ,

and the second from

Only the third inclusion is known to be s trict ([47], Lemma

p. 6 0 6 ), so $

is a plausible candidate for d

(as a ls o is

Some evidence in this direction is given by Theorem 3 ' in se ctio n 5 below. Here is a

mod p version.

R eplace

Z by

in property (i) of 3 .

Then B ro d skii’s construction works equally well to produce a quasivariety 3 p with 3 p c ® p .

T ow ers and the Magnus argum ent

4.

There is a “ stan d ard ” method of proving results about one-relator groups, which was first exploited by W. Magnus in his fundamental paper [38].

There have sin ce been various refinements (see for example [ 4 0 ] ),

but the b a sic idea remains the same. c a l trick s, it is a s follows.

Briefly, and omitting a few tech n i­

Given a one-relator group presentation

G = K with f(S1 ) C T of 7 ^ ( 0 .

K be the 2-complex

is injective.

If not, then

representing a nontrivial element

Make f “ r e a s o n a b le ,” and let K'

be a maximal tower lifting.

Then

K' is staggered, and indeed the proof

of Lemma 2 shows that we may assume of K A l s o

g(m ax a ) = x n for each 2 -c e l l a

K' is finite (for otherwise f' would factor through an in­

clu sio n , contradicting maximality), and H ^ K ' ) = 0 (for otherwise V would lift over an infinite c y c li c cover, contradictin g maximality). F in a lly , no 2 -c e l l in K' is attached by a proper power. cations of Lemma 3 show that sarily a tree), such that

K' c o ll a p s e s to a 1-complex

T

(n e c e s ­

g(y) = x n for ea ch 1-ce ll y in K - F . In

particular f ^ S 1) C g- 1 ( r ) C and

Repeated appli­

so f(S*)

ffS1) is nullhomotopic in F .

is nullhomotopic in g- 1 ( r ) ,

Contradiction.

68

5.

JAMES HOWIE

R e s u lts Many other theorems about one-relator groups can be proved using

towers in a manner similar to the proof of Theorem 11 above.

T h es e in­

clude B ro d sk ii’s Theorem 1, that torsion free one-relator groups are locally indicable.

The argument is identical, e xcep t that the d isc

replaced by some finite 2-complex

S with H1(S) = 0 (so that

D2 is

^ ( 77^ (S))

is a finitely generated, nonindicable subgroup of ^ ( K ) ). A large c l a s s of genuinely new results can be obtained by a relative version of the tower method.

I shall omit the d e ta ils, exce p t to say that 1 2 we are now considering pairs (L ,K ) of 2 -com plexes, with L = K U e U e connected, and su ch that the attaching map for e

properly involves

On the fa c e of it, there are two distinct c a s e s , depending a s

e

.

K is co n ­

nected or not, but there is in fact no e sse n tia l difference between th ese two c a s e s .

To obtain positive re su lts, we a ls o have to assume that (each

component o f ) THEOREM

2".

K has locally indicable fundamental group. With the a bove notation and c o n v en tio n s,

is in je c tiv e for e a c h com ponent

of

Theorem 2 " is just Theorem 2 when

77

^(K 1) -> 771 (L )

K. K is connected, andTheorem 2 '

otherwise.

We can a ls o obtain the following generalization of Theorem 1.

THEOREM

12 [22].

A one-relator product of lo ca lly in d ica b le groups is

lo ca lly in d ica b le, provided the relator is not a proper power (in the fr e e product). The following remark is due to S. D. Brodskii.

When solving an equa­

tion r(t) - 1 , in which r = s m is a proper power, it is enough to solv e the equation

s (t ) = 1 . Thus by Theorems 2 and 12, each nondegenerate

equation over a locally indicable group actually has a solution in some locally in d ica b le overgroup. following embedding theorem.

By a standard argument we now have the

69

ONE-RELATOR GROUP THEORY

THEOREM 3 ' [ 2 2 ] .

E v e ry lo ca lly in d ica b le group ca n be em b ed d ed in an

equationally c lo s e d lo ca lly in d ic a b le group. The next result g en eralizes a theorem of Weinbaum [66]. THEOREM 13 [22].

In a one-relator product of lo ca lly in d ic a b le groups,

no proper subw ord of the relator re p re s e n t s the identity elem en t. The Spelling Theorem of B . B . Newman [45] s a y s that , in a one-relator group with relator r = s m , any nonempty word which represents the identity element must contain a ( c y c l i c ) subword of r or r- 1

longer than

s 1* - 1 . More p recise versions are due to Gurevich [17], Schupp [55] and Pride [48].

T h e se results a ll have appropriate generalizations to one-

relator products of locally indicable groups [28].

The d etails are somewhat

te ch n ica l, but the spirit is conveyed by the following (slightly vague) statement. THEOREM 14 [28].

L e t G = ( A * B ) / N ( r ) , w here A a nd B

are locally

in d ica b le and r = s m . L e t w b e a nonempty word in A * B in N (r).

which lies

T h en eith er w is c o n ju g a t e to r or r- 1 , or w co n ta in s two

alm ost-disjoint ey e lie subw ords, e a ch of w hich is a c y c lic subw ord of r or r- 1

longer than s m_1 .

COROLLARY.

If m > 1 and A and B

have so lv a b le word problem in

T heorem 14, then G a ls o has s o lv a b le word problem . The above Corollary has been independently obtained by Brodskii and Mazurovskii (unpublished). G in the c a s e

They h a v e a ls o solved the word problem for

m = 1 , but under the str o n g e r hypothesis that A and

B

be e ffe c tiv e ly loca lly in d ica b le (there e x is t s an algorithm which, given a finite s e t of generators for a subgroup H , will decide whether or not H = { l I , and if not will exhibit an epimorphism H -> Z ). THEO REM

15 [26].

L et

G

= ( A * B ) / N ( r ) , w here A and B

indi c a b le and r = s m w here s

is not a proper power.

L et

a re lo cally KA , KB , Kc

70

JAMES HOWIE

be E ilen b erg-M a cL a n e c o m p lex es w here C

is c y c lic of order m . L e t KG = (KA v K g ) Ug Kc , w here

S = S 1 -> Ka v Kb T h en

K ( A ,1 ), K ( B , 1 ) , K (C ,1) r e s p e c t iv e ly ,

r e p re s e n ts s

KG is a K(G,

and S = S1 -> Kc

r e p re s e n ts a gen era to r.

compl ex.

Theorem 15 is a generalization of a theorem of Dyer and Vasquez [10] for one-relator groups, which can be thought of a s a geometric version of Lyndon’s Identity Theorem [34].

The usual (algebraic) version of the

Identity Theorem is in terms of the rela tio n module N(r)a ^ = N ( r )/[ N (r ), N ( r )]. C O R O LLAR Y

1.

L et G , A , B , r, s

b e a s in T heorem 1 5 .

T h en N(r)a ^

is isom orphic, a s a ZG-m odule, to the c y c lic module Z G / ( l - s ) Z G , g e n ­ era ted by r •[N(r), N ( r ) ] . Either interpretation allows one to compute the (co)homology of G in high dimensions. C O R O LLAR Y

2.

T h e re

a re natural isom orphism s, for e a c h q> 2:

Hq(G; - ) = Hq(A; - ) © Hq(B; - ) © Hq(C; - ) Hq ( ° ; - ) = Hq(A; - ) © Hq(B ; - ) © Hq(C; - ) .

Combining Corollary

2 with a result of Serre [31], we s e e that the co n ­

jugates of C = gp(s) are precisely the maximal finite subgroups of G ; and that th ese conjugates are disjoint in the strong s e n s e that C fl gCg-1 ^ {1 }'= > g f C . r\

A similar type of argument (involving computation of H

a s well as

of Hq , q > 2 ) shows that any subgroup of the form S = A fl gBg- 1 or S = A D gAg-1

(g eG -A )

or S = B fl gB g- 1

cohomological dimension at most

(g e G -B )

(geG)

has

1 , and hence [59, 63] is free.

One can

then show by induction on the length of r in the standard way that S has rank at most

1 . In the c a s e where the relator is a proper power, the

situation is much simpler.

It follows without too much difficulty from the

71

ONE-RELATOR GROUP THEORY

Spelling Theorem of [28] (s e e Theorem 1 4) that

S = ill.

Putting these

fa c t s together, we have the following result. THEOREM 16.

In the a b o v e notation, the group S is c y c lic .

If r is a

proper power then S = i l l . In the ordinary one-relator group c a s e , th e se statements are due to Bagherzadeh [4] and Newman [45] re sp ectiv ely .

6.

Open problem s I ’ll finish with some “ homework” for the interested reader.

On the

whole th e se are difficult unsolved problems, but it may be that there is hope of partial progress in some of them. 6.1 .

Does the F re i h e i ts s a tz extend to some c l a s s

greater than D ? factors

((2 = 3*3r)

(2

of groups strictly

Ideally, one would like to prove it for all torsion free . F a ilin g that, what about

Dp H Dq for distinct primes

(2 = Dp

H

DD

or C

=

p and q ?

The F r e i h e i ts s a tz fails for Dp in general b e cau se of the occurrence of torsion.

F o r example, put p = 2 ,

The c l a s s e s

fl D

p €77

"

6 .3 .

17r| > 1

(or “ m > 2 ”

77

r = a b a 2b 1 . , and are

[29].

Can Theorem 8 be improved so that “ m > 4 ”

“m >3”

B=,

are d istin ct, for fin ite s e t s of primes

contained in DD whenever 6 .2 .

A = < a ;a 4 >,

c a n be replaced by

or “ m > 1 ” )?

Is the word problem for a one-relator product of two locally indicable

groups solvable under some condition on the factors weaker than “ e f f e c ­ tively locally in d icab le” (se e se ctio n 5)? Effe ctiv e ly locally indicable clearly implies a solution to the word problem (for the fa cto rs), but this is a n e c e s s a ry condition in any c a s e , s in c e the factors embed in the whole group by the F r e i h e i ts s a tz .

72

JAMES HOWIE

Does there e x ist a locally indicable group, with solvable word problem, which is not effectively locally indicable? 6 .4 .

L et

reA*

be such that

(A * < t > ) / N ( r ) = i l l . Is A = { l ! ?

This is the so -ca lled Kervaire (or Laudenbach) conjecture.

It is c le a r

from abelian considerations that A must be perfect, and that t must occur in r with exponent sum ± 1 . It su ffices to consider the c a s e when A is an infinite simple group. 6 .5 .

Let

Does

2

2

be a system of n equations in n unknowns over a group A .

have a solution in some overgroup of A whenever det M(2) 4 0 ?-

Whenever det M(2) ^ 0 mod p for some fixed prime p ? Whenever det M(2) = ± 1 ? The third (and w eak est) of th ese would be sufficient to prove the Kervaire conjecture 6 .4 . 6.6.

Let

See [25, 60] for other related problems.

M1 C M2 be a tame embedding of ( smooth or P L ) 3-manifolds,

su ch that H2 (M2 ,M1 ;Z ) = 0 (resp. prime p ; resp.

H2 (M2 ,M1 ;Z ) = 0 for some fixed

H2 (M2 ,M1 ;Z ) = 0 for all primes

p ).

L et

S be a com­

pact orientable su rface and f : S -> M2 a (smooth or P L ) map such that f( x 2 y x “ 1y~1 , y f-> y 2zy _ 1 z “ 1 , z h> z 2x z _ 1 x ~ 1 . Then for each

n > 0 the group

Gn = < x , y , z ; f n( x ) , f n(y)> is locally indicable. Here is another way of statin g Theorem 1:

A one-relator group is

lo cally in d ica b le if and only if e a c h of its one-generator subgroups is lo ca lly in d ica ble. C O N JE C TU R E.

This s u g g ests a conjecture.

A two-relator group is locally in d ica ble if and only if

ea ch of its tw o-generator su bgroups is lo ca lly in d ica ble. Of co u rs e , one could generalize this conjecture by replacing 2

by n .

I would like to be able to prove something like this for arbitrary n , but s o far even the c a s e 6.11.

Let

n = 2

is elusive.

K be a subcomplex of a contractible 2-complex

L.

Is

^ (K )

locally indicable? This implies the Whitehead Conjecture (cf. Corollary to Theorem 6). 6 .1 2 .

Let

ZG-module

I denote the augmentation ideal in the group ring ZG . A M is p e rfe c t if IM = M . Do there e x is t nonzero, finitely

generated, perfect projective ZG-modules?

ONE-RELATOR GROUP THEORY

75

From the definition of ©-groups, it is cle a r that the answer is no whenever G no unless

(or even

G e ©p for some

p ).

Indeed, the answer is

G has a nontrivial, finitely generated, perfect subgroup [29].

On the other hand, if G does contain such a subgroup, then there e x is t s an infinitely g e n e ra te d perfect projective ZG-module [68].

J A M E S H O W IE D E P A R T M E N T O F M A T H E M A T IC S U N IV E R S IT Y O F G L A S G O W U N IV E R S IT Y G A R D E N S G L A S G O W G 12 8QW U. K.

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B. Baum slag and S. J . Pride, An e xten sion of the F r e i h e i ts s a tz , Math. P ro c. Camb. Phil. Soc. 8 9 ( 1 9 8 1 ) , 3 5 -4 1 .

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G. H. Bagherzadeh, Commutativity in one-relator groups, J . London Math. Soc. 1 3 ( 1 9 7 6 ) , 4 5 9 -4 7 1 .

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[9]

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[10]

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76

JAMES HOWIE

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F . Gonzalez-Acuna and H. Short, Knot surgery and primeness, Math. P roc. Comb. Phil. Soc. 9 9 ( 1 9 8 6 ) , 8 9 -1 0 2.

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, Units in group rings, D. Phil. T h e s i s , University of Oxford 19 40.

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[28] J . Howie and S. J . Pride, Spelling theorems for staggered generalized 2 -com plexes, with applications, Invent. Math. 7 6 ( 1 9 8 4 ) , 5 5-74 . [29] J . Howie and H . R . Schneebeli, Homological and topological proper­ ties of locally indicable groups, Manuscripta Math. 4 4 ( 1 9 8 3 ) , 7 1 -9 3 . [30] J . Howie and H. Short, The band sum problem, J . London Math. Soc. 31 (19 8 5 ), 5 7 1 -5 7 6 . [31] J . Huebschmann, Cohomology theory of asp h erica l groups and small can cellatio n groups, J . Pure Appl. Alg. 1 4 ( 1 9 7 9 ) , 1 3 7-14 3.

77

ONE-RELATOR GROUP THEORY

[32] L . Y a . Leifman and D . L . Johnson (e d s .) The Kourovka Notebook. Unsolved problems in group theory, AMS T ranslations (2) 1 2 1, 19 83 . [33] F . Levin, Solutions of equations over groups, Bull. Amer. Math. Soc. 6 8 ( 1 9 6 2 ) , 6 0 3 -6 0 4 . [34] R . C . Lyndon, Cohomology theory of groups with a single defining relation, Ann. of Math. 5 2 ( 1 9 5 0 ) , 6 5 0 -6 6 5 . [35] __________ , On the F r e i h e i t s s a t z , J . London Math. Soc. 5 ( 1 9 7 2 ) , 9 5 -10 1. [36] __________ , Equations in groups, Bol. Soc. B ra s. Math. 11 (1 9 80 ), 7 9 -1 0 2 . [37] R . C . Lyndon and P . E . Schupp, Combinatorial Group Theory, Springer-Verlag 1 97 7 . [38] W. Magnus, Ueber diskontinuierliche Gruppen mit einer definierenden R elation (Der F r e i h e i t s s a t z ) , J . Reine Angew. Math. 1 6 3 ( 1 9 3 0 ) , 1 4 1 -1 6 5 . [39] A . I . Mal’c e v , Algebraic System s, Springer-Verlag 1 9 7 3 . [40] J . McCool and P . E . Schupp, On one-relator groups and HNN e x te n ­ sio n s , J . Austral. Math. Soc. 1 6 ( 1 9 7 3 ) , 2 4 9 -2 5 6 . [41] J . Milnor, On the three-dimensional Brieskorn manifolds Ann. of Math. Studies 8 4 ( 1 9 7 5 ) , 1 7 5 -2 2 5 .

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JAMES HOWIE

[52] H. Schiek, Adjunktionsproblem und inkompressible Relationen I, Math. Ann. 1 4 6 ( 1 9 6 2 ) , 3 1 4 -3 2 0 ; II, Math. Ann. 161 (1 9 6 5 ), 16 3 -1 7 0 . [53]

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__________ , Surfaces in three-manifolds and non-singular equations in groups, Math. Z . 1 8 4 ( 1 9 8 3 ) , 1-17.

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GRAPHICAL TH EO R Y OF AUTOMORPHISMS OF F R E E GROUPS John R. Stallin g s*

Abstract.

T h e e q u a liz e r o f tw o m o n o m o rp h is m s o f f i n i t e l y g e n e ra te d fre e g ro u p s

is f i n i t e l y g e n e ra te d .

T h is g e n e r a liz e s G e r s te n ’ s T h e o re m th a t th e f ix e d s u b g ro u p

o f a n a u to m o rp h is m o f a f i n i t e l y g e n e ra te d fre e g ro u p is f i n i t e l y g e n e ra te d . p r o o f is a n e x p o s it io n o f G e r s te n 's id e a s , u s in g g ra p h - th e o ry . d e v e lo p e d w h ic h m a y h a v e o th e r u s e s , ‘ ‘ d y a d s , ”

The

T e c h n iq u e s a re

“ f o l d s , ” “ la d d e r-a tta c h m e n ts . ”

T h e p a p e r c o n ta in s a h i s t o r ic a l s k e tc h a n d a l i s t o f r e la te d u n s o lv e d p ro b le m s , a n d a ls o a n a p p e n d ix r e la t in g d y a d s to H e e g a a rd d ia g ra m s o f 3 - m a n ifo ld s .

Introduction In the fall of 1 9 8 2 , S.M. Gersten [G3] proved his fixed-point theorem: If a : F -> F

G E R S T E N ’S T h e o r e m .

is an autom orphism of a fin itely

g e n era ted fr e e group, then F i x (a) = lw A . Three important properties are 1.

An immersion f : T ^ A is injective on

2.

A fold map e : V -> T / ( e 1 = ^ 2 ) is both su rjective itse lf, and is

su rjectiv e on 3.

.

.

A non-degenerate map f is an immersion if and only if it admits

no fold. 2.4 . paths

p

Here and q

are some tec h n ical definitions about paths. in F , that

We sa y of

p is a left seg m en t of q , and write

p < q , if : ‘( p ) = i ( q ) ; If p

p = X j - - - x n , q = y 1 ---yk ; n < k ; a n d

and q

V i e [l,n ],

Xj = yj .

are paths with the same initial vertex, we define pAq

to be the unique maximal common left seg m en t of p and q . Note th ese f a c t s : If f *.r -> A

is a map of graphs and

p and

q are paths in F

the same initial vertex, then f(p Aq) < f(p)Af(q) .

If, additionally, In any c a s e ,

p

f is an immersion, then f(p Aq) = f(p )A f(q ).

0 / which is an identification of a pair of edges

, e2

having the same initial vertex and having the properties that a ( e j ) = a ( e 2 ) and

Then a

= j8(e2 ) .

and (3' are the unique graph-maps such

that a cf) = a and /3' = /3 . We c a l l

(a ,/3 ,0 )

d eg en era te-im m ersiv e or “ D I ,” if no fold is p o s s i­

ble; that is, if a|Dj8 and j3|Da are immersions. A fold reduces the minor complexity and is surjectiv e on

. Thus

we obtain: 3.4.

LEM M A.

L et

(a ,/3 , 0 )

b e a (T ,A )-d y a d .

T h en there is a

(F,A)-c/yac/ (a ',j3 ',® ') and a graph-map . ® ^ ® ' su ch that (i)

a eft = a , f3'cf> = /3 .

In other w ords, cf> is a map of dyads.

(ii)

cf> is the com position of a fin ite number of fo ld s.

(iii)

(a ',f3 ',® ')

is d eg en era te-im m ersiv e. (A nd it follow s from (ii) that: )

4.

(iv)

cf> is s u rje c tiv e .

(v)

cf)^ : tt^( 0 ) -> 7r1 (0 O

is s u rje c tiv e .

Q uasi-im m ersion 4 .1 .

(/3 ,a ,0 ).

A (r ,A )-d y a d

(a, j3 , 0 )

is said to be symmetric to the (A ,r)-d y a d

In what follows, all definitions and theorems are to be inter­

88

JOHN R. STALLINGS

preted symmetrically.

F o r example, “ q uasi-fold ’ ’ is defined in terms of

the ordering (a ,/3 ) and is supposed to include the symmetric possibility. The major co m p lexity of a (r ,A )-d y a d is the sum of the number of components of Da and the number of components of D/3 . 4 .2 .

A fo ld of a dyad d o es not in c re a s e the major

P R O P O S IT IO N .

com plexity. Proof.

What a fold does is to perform a fold on one of Da , D/3 and to

identify a pair of v e rtic e s , which may already be equal, of the other.

If

Da is folded, its number of components s ta y s the same, and the number of components of Dj3 s ta y s the same or d e c re a s e s by one. □ 4 .3 .

A q u asi-fo ld of a dyad

counterpart):

A triple

(e^ e^ p )

(a ,/3 ,0 )

is the following (or its symmetric

where

and e

edges of D/3 such that a ^ ) = a ( e 2 ) ; where in Da from

to

are non-degenerate

p is a non-degenerate path

t(e2 ) ; such that r ( e 1) and r ( e 2 ) are in different

components of Da . A dyad is said to be qua si-im m ersiv e or “ Q l” if it has no quasi-folds. If a dyad is both DI and QI we ca ll it “ DQI.” 4 .4 .

A ladder L = L (a ,p )

in ( r , A )

is the following, or its symmetric

counterpart: L

is a (r ,A )-d y a d ;

a

is a non-degenerate edge of T , and

p is a

non-degenerate path in A of length n > 0 . The underlying graph of L c o n s is ts of two disjoint a rcs

A^ , A 2 sub­

divided into n = |p| ed ges; th ese are the s id e s of the ladder; and of n+1 edges

^0 , £1 ,-*-,£n connectin g corresponding v ertices in A^^ and A 2 ,

with initial v ertices in A 1 ; th ese are the ru n gs of the ladder. Here are the two graph maps a , /3 : L -> T , A . a A 1 and A 2 and maps each

£■ to a .

the path p and is degenerate on the (Figure 1).

is degenerate on both

/3 maps each of A 1 and A 2 by £• . We ca n describe this pictorially

To describe a dyad in pictures, we draw a picture of a directed

AUTOMORPHISMS

89

graph, and label each edge with a pair of lab els, the first describing the map a , the second d e s c ribing the map /3 ; the symbol 0 denotes the unique vertex of T

or A ; an edge labelled

“ cO” then belongs to D/3

and an edge labelled “ 0 z M belongs to Da .

OU

ov F ig u r e 1

Note that both a and (3 map ^ ( L ) , with any basepoint, to ! l S , that is, trivially. 4 .5 .

If the (r ,A )-d y a d

co n stru ct the ladder

(a ,/3 ,0 )

A 1 to 0

rungs

en to e^ and e^

dyad

(e^e

,p ) , we

L ( a ( e t ), /3(p)) and attach it to ( a , / 3 , 0 ) , by identi­

fying the side and

admits a quasi-fold

along the path p and identifying the extreme resp e ctive ly .

This results in a new

(a', f3', 0 ' ) . We can imagine this attachment done in s t a g e s ; first

identify the corner vertex

i(£Q) to

t ( p ) ; then fold, in s u c c e s s i o n , the

edges of A 1 onto the edges of p ; and then fold the extreme rungs onto e 1 and e 2 . 4 .6 .

P R O P O S IT IO N .

T h e attachm ent of a ladder L

to a dyad ( a , / 3 , 0 )

along a qu a si-fo ld re d u c e s the major co m p lexity . Proof.

D/3 changes by adding the rungs of the ladder and the ve rtice s of

A^ ; s in c e one vertex of each rung is in D/3 to begin with, the number of components of D/3 does not change.

But Da

is changed by joining the

90

JOHN R. STALLINGS

points r ( e 1) and r ( e 2 ) by the arc

A 2 , thus reducing the number of com­

ponents of Da by 1 . □ 4 .7 .

LEM M A.

If ( a , / 3 , @ )

is any (Y ,k )-d y a d , then th ere is a ( r , A ) -

dyad (a ',/3 ',® ') and a map ® -> ® ' s u c h that 1.

a'cf) = a , /3'c/>=/3.

2.

^

is the com position of a fin ite num ber of fo ld s and ladder

attachm ents. 3. Proof.

( a ',P ',® ')

i s DQI .

If the dyad is not QI, a quasi-fold may involve a path p of length

0 ; in this c a s e , the dyad may be folded to reduce the major complexity; if the path

p has length greater than 0 , a ladder may be attached to re ­

duce the major complexity.

We can then reduce the minor complexity by

folding, and this does not, by 4 . 2 , in crea se the major complexity.

Thus, a

double induction e sta b lis h e s that we eventually reach the point that there are neither quasi-folds nor folds. □

5.

T h e com m on-kernel condition 5 .1 .

P R O P O S IT IO N .

L et 0 : T - > T ' ,

a:T^C,

j3:T^D,

a ':T '-> C ,

/3': T '-* D b e group-hom omorphisms su c h that a - a cf), f3 = fi'cf). S u p p o se ker a = ker /3 . (i)

T h en

If cf) is s u rje c tiv e , then ker a = ker (3 '.

(ii) If T '= T * K ,

a fr e e product, 0 : T -» T * K is the in clu sio n , and

a '( K ) = 111 = /3 '( K ) , then k e r a ' = k e r / 3 \ P roof. 5 .2 .

Easy. □ L et

(a ,/3 ,0 )

be a (F,A )-dyad.

We sa y that this dyad s a ti s fie s

the com m on-kernel condition if ker a = ker /3 on tt^ with some ch o ice of

AUTOMORPHISMS

basepoint in © .

Since 0

91

is con nected , this is independent of the ch o ice

of basepoint. 5 .3 .

If ( a , / 3 , 0 )

LEM M A.

is a ( r , A )-dyad with common k e rn e l} then

(by 4 .7 ) there is a DQI-dyad ( a , f i ',® ') and a map of dyads where

We have y 8 ( f ) = )8(f2 ) and

i(fj) is joined to

in D/3 . Therefore, by the symmetric version of the above

r(fj)

a ( e 3 ) = a ( e 1) .

L e t q = ^ q ' and

/3(q)

is joined to r(f2 ) by an edge By induction on

e” 1 in D/3 such that

|p| , we se e that e^ and e 2 are extreme

rungs of a ladder mapped into ® ; by adjoining the extra square ! e 1 ,e ^ ,f1 ,f2 ! we get e 1 and e 2 to be the extreme rungs of the desired ladder. □ 5 .7 .

REMARK.

The above, 5 .6 , if the form of G ersten’s “ path-

su rge ry” axiom G4 which is appropriate in this situation. 5.8.

In a (r,A )-d y a d

( a , / 3 , 0 ) , a non-degenerate path p is called

(a ,fi)-re d u c e d if #a(p) and #/3(p) are reduced paths in T and A . (R e ca ll that #q is got from q by striking out degenerate e d g e s .) P s e u d o-homotopy is the equivalence relation

^

on paths in ©

generated by two operations: (a) Homotopy:

pee- 1 q ^ pq , and pvq — pq if v is a vertex.

(b) Replacing a segment joining diagonal points of a ladder by the other segment; that is: pefq ^ pfre'q when f ' e T - 1 e _1

is a clo se d path,

e , e '< r D /3 , f, f' ] = [a (q )],

[/8(p)] - [/3(q)L

93

AUTOMORPHISMS

Proof.

Elementary. □

5 .1 0 .

P R O P O S IT IO N .

mon k ern el.

L et ( a ,/ 3 ,0 )

be a DQI (T ,k y d y a d with co m ­

For ev ery path p in ® , th ere e x is t s an (a, fi)-r e d u c e d path

p ' s u c h that p

p'; and in this c a s e Ip '! = |[a(p)]| + |[/3(p)]| .

P roof.

C hoose

p ' — p so that

|p'| is minimal.

(a,13)-reduced, it would have a segment s

If p ' were not

of the following kind or its

symmetry: p ' = P l s p 2 , s = e ~ 1 p 3e 2 where e 1 , e 2 are parallel edges in D/3 , joined by a path p3 in Da . Since in 0

|p'| is minimal,

p3 is a reduced path.

By 5 .6 , there is a ladder

connecting th e se e d g e s, and so there is a path p4 in Da from

r ( e 1) to r ( e 2 ) su ch that j3(p3 ) = /3(p4) . Then:

P" =

P i P 4 P2

~

P'

and p" would have shorter length. □ 5 .1 1 . k ern el.

THEO REM .

L et

initial v ertex.

L et ( a ,/ 3 ,0 )

be a DQI (T,/\)-dyad with common

p and q be (a ,fi)-re d u c e d paths in 0 C o n sid er paths

p ' , q ' w hich a re (a fj3)-red u ced , s u c h that

p Aq < p'Aq' and p' — p an d q' ^ q . s u c h that |p/ Aq, | is maximal.

having the sam e

Of th e s e , c h o o se a pair p ', q /

T h en

# a ( p ^ q ' ) = #a(p ) a #a(q') , and #j8(p'Aq') - #jS(p') a #j8(q') • P roof.

Note that

such a pair with

|p'Aq'| jp'Aq'j

is bounded by maximal e x i s t s .

]p'| = |[a(p)]| + |[/3(p)]| . So Let

r = p ' a q'.

C learly

94

JOHN R. STALLINGS

Suppose #a( r) < #a(p ') a #a(q')*

Let

e x be the first edge of D/3 in

p' after r , and e 2 the first edge of D/3 in q' after r . The situation is that

p' = r s ^ j t j , q ' = r s 2 e 2t 2

where Sj

By 5 .6 , (a ,/3 ,0 ).

and s 2 are paths in D a , and a ( e 1) = a ( e 2) .

The picture is:

e ^ and e 2 are joined by a ladder L ( a ( e 1) ,/ 3 ( s ~ 1s 2 )) in Thus, a s in the p ictu re , there is an edge e 3 in D/3 with

a ( e 3) = a ( e 1 ) , and paths jS(s2 ) = /S(s2 ^).

s ^ s^

in Da with /3 ( s x) = /^ ( s ^ ) and

Then p^= r e 3s 1 ^t1 , and q ^ = r e 3s 2 ' t 2

are pseudo-homotopic to p' and q ' , but they have

r e 3 < p " a q" and so

|p'a q'| was not maximal. □ 5.12.

C O R O LLAR Y.

with common k ern el.

L et

(a ,/3 ,0 )

be a

r,& )-d y a d w hich is DQI

L e t p and q b e paths in 0

su c h that p an d q

have the sam e initial v ertex and s u ch that [a(p)] = [a ( q ) ] , [/3(p)] = [/3(q )]. T h en p

q . In particular, p and q have the sam e terminal v ertex .

95

AUTOMORPHISMS

Proof.

Apply 5 .1 1 to find (a,/3)-reduced

q ^ q ', and the co n clu sio n s of 5 .11 hold. the paths

p ', q ' such that p ', q'

Since

# a ( p ' ) , e t c . , are reduced paths in T

p

^ p ' , and

are (a,/3)-reduced

and A .

Thus the

assumption that the homotopy c l a s s e s of the images of p and q under a and /3 are equal implies # a (p 'Aq ') = #a( p') = #a( q ') # ^ (p 'A q ') = # 0 ( p ') = #/3(q') . We can thus compute the lengths of p ', q ', and they are all the same. 5 .1 3 .

REMARK.

Thus

p 'a

q ' ,and

find that

p ' a q' = p ' = q '. □

This implies that the kernel of the homomorphism « * X /3* : ^ l( ® )

is normally generated by ladders.

x 77j (A)

This is not surprising, s in c e this

image is isomorphic to the image of a , which is finitely generated and free, and thus finitely related. squares.

The ‘ ‘ re la tio n s ’ ’ are given by ladder-

The hope that this could be arranged only under the assumption

that the image of a^ x

is finitely related (without assuming the com­

mon kernel condition) is the b a sis for problem P 6 .

6.

C o in c id e n c e s We sh all now d is c u s s the c a s e of ( r , r ) - d y a d s .

That is ,

A = F.

In

this c a s e , we can compare a(p) and /S(p) and therefore d is c u s s the equalizer on 6.1.

.

L e t a : T -» C , /3 : T -> C be homomorphisms of groups.

We

define the im age of the e q u a liz e r : IEq(«,/3) = {x C

L et T * K is the

in clu sio n , and a \ K ) = ! l j =

j8'(K ), ^e/7 IEq(a',|8') = IE Q (a,/3). Proof.

Easy. □

6 .3 .

LEM M A.

//

is a ( F ,T )-d y a d with common k ern el, then

( a , / 3 ,0 )

there is a DQI-dyad (a', fi', 0 ' )

and a map of dyads cf> : 0

0'

w hich is

the com position of a fin ite s e r i e s of fo ld s and ladder-attachm ent s. end resu lt (a', fd', 0 ' )

The

s a tis fie s the com m on-kernel condition, and I E q ( a ; , £ * ) = IE q C a *,^ ) .

[This is to be interpreted by choosin g a basepoint in 0 0 ' is to be the basepoint of 0 ' . on

Then

whose image in

, e t c . , are the homomorphisms

relative to th ese basepo ints.]

Proof. This is a restatement of 5.3 with the additional conclusion about IEq, which follows from 6 .2 s in ce folds and ladder-attachments satisfy on the conditions of 6 .2 (i) and (ii). □ 6 .4 .

Suppose that

(a ,/3 ,0 )

is a DQI ( r , F ) - d y a d with common kernel.

An (a,/3)-reduced path p in 0

will be called invariant (or a

c o in c i d e n c e ) if #ct(p) = #/3(p) . L e t e be an edge of 0 .

There may or may not be an invariant path

p whose first edge is e . [ R E M A R K :

As Gersten shows, there is an

efficient and direct mechanical procedure to find such a path exists.

p if it

This procedure may not terminate if such a path does not e x i s t .]

If there is an invariant path p whose first edge is

e , then out of all

su ch , ch o o se one whose length is minimal, and c a ll it

m ( e ) . The plan

now is to invent a graph made up of |m(e)i which determines, up to pseudo-homotopy, all the invariant paths.

AUTOMORPHISMS

6.5.

Let

(a ,/3 ,0 )

97

be a DQI ( r , r ) - d y a d with common kernel, and

c h o o s e , a s above, for e a c h edge e

of 0

for which it is p ossib le, a

minimal invariant path m ( e ) . Define the c o in c id e n c e graph an d map,

(3, o) ,

a s fo llo w s: \A (e)S , where A (e)

F i r s t , we take disjoint a r c s

|m(e)| e d g e s, and the s e t of v e rtices terminal v e rtice s of A(e) to the graph

6 .6 .

V ( 0 ) ; identify the initial and

t(m(e)) and r(m (e)), re sp ectiv ely.

o \3 -> 0

On A ( e ) , let THEO REM .

as follows:

On V ( 0 ) , let

L et (a ,/3 ,0 )

b e a DQI (V ,T )-d y a d with common

(3, o)

w hich the c o in c id e n c e graph and map (i) For ev ery path p of & V (@ )>

o be the

o define the path m ( e ) .

k e rn el, and let a c h o ic e of minimal invariant paths im(e)i

in

This is

3.

Then, we define identity.

is subdivided into

a re form ed.

b e m ade, with T h en :

w hose initial and terminal v e rt ic e s a re

r

r

[ a (a (p ))] = [/3(a(p))] .

(ii) For ev ery path q of 0 , path p of a

if [a(q)] = [/3 (q )], then there e x is t s a

w h o se endpoints a re those of q in V ( 0 ) , s u c h that #cr(p) — #q .

Proof,

(i) Since

p has its endpoints in V ( 0 ) , it is homotopic to a

product of paths, ea ch cro ssin g some A(e) once, and so

C

a re m onom orphism s, then Eq(a,/3)

is

fin itely g en era ted . Proof.

Note that

ker a = ker /3 = l l i , and IEq(a,/3) = a (E q (a ,/3 ))

is

isomorphic to E q ( a , / 3 ) . D 7 .3 .

REMARK.

7 .2 implies 7.1 by an e a s y argument, left to the

reader.

But 7.1 is the form of the result which is proved by this graph -

theory.

The monomorphism condition deteriorates into the common kernel

condition b e c a u s e of the need to a tta c h ladders.

8.

Two exa m p les

F ig u r e 4

F igure 3

100

JOHN R. STALLINGS

T h e se are DQI ( r , r ) - d y a d s , with the pictorial description d is cu s se d in 4.4 .

The circled v e rtic e s are the basepoints.

Figure 4 is a general c a s e ,

in which various types of ladders can be made out.

Figure 3 is a picture

of one of G ersten ’s “ CMT” automorphisms. What is notable about Figure 3 is that there is a minimal invariant path starting at the basepoint (it happens to be a c lo se d path); it has length 10 0.

T hus, long after you

want to quit, if you keep at it, you may find that a minimal invariant path exists.

9.

A p p en d ix : H eeg a a rd diagram s 9 .1 .

The notions of “ d y a d ,” “ f o ld ,” and “ degenerate-immersive” can

be used to explain some of the work of Volodin, Kuznetsov, and Fomenko [V-K-F].

They developed a technique for simplifying Heegaard diagrams

of 3-manifolds.

They conjectured that an unsimplifiable diagram of the

3-sphere was the standard one, and got a computer to verify the conjecture on 1 ,0 0 0 , 0 0 0 randomly constructed diagrams.

Viro and Kobelskii [V-K]

then drew a picture of a genus three counterexample.

This is an in stance

in mathematics of the common “ s c i e n t i f i c ” phenomenon of a survey of an unexpectedly biased population (cf. [ L a ] ) .

N ev erth eless, the technique of

Volodin et al. , is worth further study. 9 .2 .

A H eega a rd diagram of genus

connected, oriented 2-manifold

T

n , ( T , A , B ) , c o n s is ts of a clo se d ,

of genus

n , and two s e t s ,

A and B ,

each co n sistin g of n oriented disjoint simple clo se d curves on T , such that

T - A and

T - B are connected (and therefore of genus

such that A and struct Ha

B

intersect transversely.

by attaching

3 -c e l l to the result;

Given ( T , A , B ) , we can c o n ­

n 2 -c e l ls to T along A , and then attaching a

HA is a handlebody, the resultof taking the 3 -c e l l

and identifying n pairs of disjoint 2 - c e l ls on its boundary. there is the handlebody common boundary and every such

T

0 ), and

Similarly,

HB . The union of HA and HB along their

is a c lo s e d , connected, orientable 3-manifold

M;

M is obtained from some Heegaard diagram in this way.

101

AUTOMORPHISMS

We obtain the same 3-manifold M if certain ch an g es are made on A and B . Thu s, we ca n change intersection of A f l B .

A by an isotopy to minimize the points of

If some component of T - A U B

connected, then an a n a ly sis shows that

is not simply

M can be decomposed into a

non-trivial connected sum, e a c h summand having a Heegaard diagram of smaller genus (or, in the extreme c a s e , we have = 1 ).

M = S 1 x S 2 and genus

Once we have made th ese immediately simplifying ch a n g e s, we

sh all have what we c a l l a sim p le Heegaard diagram: ponents of T - A U B A flB

One having all com­

simply con nected , and having the cardinality of

minimal within the isotopy c l a s s of A on T .

9.3 .

A su b sp a ce

A of a topological s p a c e

X

is said to be

b ico lla red in X , if it has a neighborhood homeomorphic to ( - 1 , + 1 ) x A in su ch a way that A corresponds to 0 x A . In a Heegaard diagram and B

( T ,A ,B ) we s e e that A is bicollared in T ,

is bicollared in T , and AUB - AflB

9 .4 .

Suppose A is bicollared in X .

is bicollared in T - AHB .

We co n stru ct a graph F = F ( X , A )

a s follows: V (r)

is the se t of connected components of X - A .

E (r)

con tain s a reverse pair of edges for e ach component

think of e e E ( F )

of A .We

as a “ normal orientation’ ’ to the corresponding com­

ponent of A , and thus c o n n e cts one side of that component to the other sid e.

t(e)

The components of X - A containing th ese sid es constitute

and r ( e ) . 9 .6 . struct

r

Let = r(T,A),

(T ,A , B ) be a Heegaard diagram of genus

ed ges.

r

We can c o n ­

A = T ( T , B ) , and © = T ( T - A f l B , A U B - A f l B ) .

There are maps of graphs in the Gersten s e n s e , Both

n.

a : © -> T , /8 :

© -> A.

and A are 1 -vertex graphs, ea ch with n reverse

An edge of © corresponding to a component of A - AflB

B - AflB ) belongs to D/8 (resp.,

Da ).

Thus

(a ,/8 ,@ )

pairs of (r esp.,

is a (r ,A )-d y a d .

JOHN R. STALLINGS

102

(A similar construction works for any s p a c e

X and pair of bicollared

su b s p a ces in tersecting tran sv e rsely .) Now, there is an embedding of the topological realization of the a s s o c ia t e d graph ® into T . Surrounding each point of intersection of A and

B there is the image of a ladder-square.

and of B

E a c h component of A

gives rise to a ladder that goes around in a c ir c le .

If (T ,A , B )

is a simple Heegaard diagram, then

Da

and

D/3

are

deformation retracts of T - A and T - B , resp ectively. 9 .7 .

What does it mean for the dyad

Heegaard diagram ( T , A ,B ) to admit a fold?

(a ,/3 ,® )

a s s o c ia te d to the

It is the following, or its

symmetric an alogue:

There is a component E

of T - AUB

on whose boundary there are

two components of A - AflB which belong to the same component A^

of

A , and which are oriented coherently in the boundary of E . Volodin et a l. , term the arc

y which is drawn in Figure 5, a “ w a v e .”

Ordinarily, in higher dimensions, a fold would be topologically realized by joining up th ese two p ieces of Bd E y in the picture.

by a tube along the arc

However, b eca u se of the fact that T

is 2-dimensional,

this com plicates the picture by replacing one component A 1 of A by

AUTOMORPHISMS

two cu rves

A 11

and

103

A 12 . Thus, the attempt to simplify, in fa ct com ­

p lic a te s ; this is a common exp erience in 3-manifold labor. 9.8 . one of

H ow ever, Volodin et a l .t note that one can simply throw away or A 12 . This produces a new Heegaard diagram

(T ,A ',B )

of the same 3-manifold, with fewer points of intersection A ' f l B .

On the

a s s o c ia t e d dyad this produces a change that is rather different from simply a fold; in particular, the graph F

ca n be imagined to change,

changing the free b a sis of n 1 ( T ) = 7Jp1(Ha ) . The ch o ice of which of A 11

or A 12

to omit is made like th is:

A " = A - A j ; then T - A " is a surface of genus H ^ T - A", B d(T - A " )) . Homologically, l e a s t one of A ^ other one away.

or A 12

is

1 , and A^ 4 0 in

A 1 = A 1 1 + A 1 2 , and therefore at

4 0 in H j ( T - A", B d ( T - A " )) ; throw the

This c h o ic e insures the condition that T - A ' be c o n ­

nected and thus be of genus 9 .9 .

Let

0.

The co n jectu re of Volodin et al . , which was disproved by the

Viro-Kobelskii example, was that if ( T ,A ,B )

is a simple Heegaard

diagram of the 3-sphere whose a s s o c ia t e d dyad admits no fold, then it is the obvious diagram of genus

0.

In other words, given any Heegaard

diagram of S 3 in which the number of components of A f l B

has been

minimized, then either T - AUB has a non-s imply-connected piece, or e ls e there is a “ w a v e .” If that conjecture had been true, it would have given some hope that the P o in care Conjecture might be attacked from that angle.

There are

other kinds of manipulations on Heegaard diagrams; the dyad picture gives an a b s tra c t idea of what is happening to the fundamental groups. It might be worthwhile to pursue this further to find some clarificatio n of algorithmic problems in 3-manifolds, a s well a s to quest for the solution to the Poin care C onjectu re. Perhaps the major te ch n ic a l snag is that a path-surgery result, such as 5 .6 , is lacking. kernel condition.

The dyads that come up do not have the common-

104

JOHN R. STALLINGS

D E P A R T M E N T O F M A T H E M A T IC S U N IV E R S IT Y O F C A L IF O R N IA B E R K E L E Y , C A L IF O R N IA 9 4 7 2 0

REFERENCES [Co]

D. Cooper, “ Automorphisms of F r e e Groups Have F . G. F ix e d Point S e t s , ” Preprint Prin ceton U. (1983).

[C-V]

M. Culler, K. Vogtmann, “ Moduli of Graphs and Automorphisms of F r e e G roup s,” Inv. math (to appear).

[D-S]

J . L . Dyer, G. P . Scott, “ Periodic Automorphisms of F r e e Groups,” Comm. Al g. 3 ( 1 9 7 5 ) , 1 9 5 -2 0 1 .

[G l]

S. M. Gersten, “ On F ixe d Poin ts of Certain Automorphisms of F r e e G roups,” P ro c. L ondon Math. S o c. 4 8 ( 1 9 8 4 ) , 7 2 -90 , “ Addendum,” 4 9 ( 1 9 8 4 ) , 3 4 0 -3 4 2 .

[G2]

, “ On F ixe d Points of Automorphisms of F in itely Generated F r e e G roups,” B u ll. A m er. Math. S o c. 8 ( 1 9 8 3 ) , 4 5 1 -4 5 4 .

[G3]

, “ F ix ed Points of Automorphisms of F r e e Groups,” Adv. in Math, (to appear).

[G4]

, T opology of the Autom orphism Group of a F r e e Group, (to appear).

[G-T]

R . Z . Goldstein, E . C . Turner, “ Automorphisms of F r e e Groups and Their F ix e d P o i n t s , ” Inv. math. 7 8 ( 1 9 8 4 ) , 1-12.

[Hoa]

A. H. M. Hoare, “ On Automorphisms of F r e e Groups” (to appear).

[How]

A. G. Howson, “ On the In tersection of F in itely Generated F r e e G roups,” J . L ondon Math. S o c. 2 9 ( 1 9 5 4 ) , 4 2 8 -4 3 4 .

[j-S ]

W. J a c o , P . B . Shalen, “ Surface Homeomorphisms and P e r io d ic i ty ,” Topology 1 6 ( 1 9 7 7 ) , 3 4 7 -3 6 7 .

[L a]

S. Lang, T h e F ile , Springer-Verlag (1 9 8 1).

[L-S]

R . C . Lyndon, P . E . Schupp, Com binatorial Group Theory, Springer-Verlag (1977).

[Me]

J . McCool, “ Some Finitely Presented Subgroups of the Automor­ phism Group of a F r e e Group,” J . A lgeb ra 3 5 ( 1 9 7 5 ) , 2 0 5 -2 1 3 .

[Mi]

K. A. Mihailova, “ The Occurrence Problem for Direct Products of G roups,” Doklady A kad. Nauk SSSR 1 1 9 ( 1 9 5 8 ) , 1 1 0 3 -1 1 0 5 .

[Ni]

J . Nielsen, “ Die Isomorphismengruppe der F re ie n Gruppen,” Math. A nn. 9 1 ( 1 9 2 4 ) , 169-2 09.

[Po]

E . L . P o s t, “ A Variant of a R ecursively Unsolvable P rob lem ,” B u ll. A m er. Math. S o c. 5 2 ( 1 9 4 6 ) , 2 6 4 -2 6 8 .

AUTOMORPHISMS

[Fa]

105

E . S . Rapaport, “ On F r e e Groups and their Automorphisms,” A cta Math. 9 9 ( 1 9 5 8 ) , 1 3 9 -1 6 3 .

[Se]

J - P . Serre, A rb res , A m algam es SL„ , Asterisque 4 6 , Soc. Math, de F r a n c e (1 97 7 ).

[Sq]

C. Squier (private communication).

[S tl]

J . R . Stallings, “ Topologically Unrealizable Automorphisms of F r e e G roups,” P ro c. A m er. Math. S o c. 8 4 ( 1 9 8 2 ) , 2 1 -2 4 .

[St2]

----------------, “ Topology of F in ite G ra p h s,” Inv. math. 7 1 ( 1 9 8 3 ) , 5 5 1 -5 6 5 .

[V-K]

0 . Y a . Viro, V. L . Kobelskii, “ The Volod in-Kuznetsov-Fomenko Hypothesis on Heegaard Diagrams of the 3-Sphere is F a l s e , ” U spehi Mat. Nauk 3 2 :5 (197) (1 9 7 7 ), 1 7 5 -1 7 6 .

[V -K-F]

1. A. Volodin, V . E . Kuznetsov, A . T . Fomenko, “ On the Problem of Algorithmic Discrimination of the Standard Three-Dimensional S p h ere,” U sp eh i Mat. Nauk 2 9 :5 (179) (1 9 74 ), 7 1 -1 6 8 . English translation : R u ssia n Math. S u rv ey s 2 9 :5 , London Math. Soc. (1 9 7 4 ) , 7 1 -1 7 2 .

[Whl]

J . H . C . Whitehead, “ On Certain Sets of Elements in a F r e e Group,” P ro c. London Math. S o c. 4 1 ( 1 9 3 6 ) , 4 8 -5 6 .

[Wh2]

--------------- , “ On Equivalent Sets of Elements in a F r e e Group,” A nn. of Math. 3 7 ( 1 9 3 6 ) , 7 8 2 -8 0 0 .

P E A K REDUCTION AND AUTOMORPHISMS OF F R E E GROUPS AND F R E E PRODUCTS Donald J . Collins “ Every mountain and hill shall be made low, the crooked made straight and the rough p lace s sm oo th .”

Abstract.

(Isaiah 4 0 :4 ) .

In a w e l l k n o w n p a p e r p u b lis h e d in 1 9 3 6 , J . H . C . W h ite h e a d g a v e a n

a lg o r it h m to d e c id e w h e th e r tw o e le m e n ts o f a fre e g ro u p o f f i n i t e ra n k a re e q u iv a ­ le n t u n d e r a n a u to m o rp h is m . A n a lg e b r a ic p r o o f o f W h ite h e a d ’ s r e s u l t w a s la t e r g iv e n b y P . J . H ig g in s a n d R . C . L y n d o n a n d t h i s w a s th e b a s is fo r th re e im p o r ta n t p a p e rs b y J . M c C o o l on p r e s e n t a tio n s o f g ro u p s o f a u to m o rp h is m s o f a fre e g ro u p . T h e p u rp o s e o f th is a r t ic l e is to d e s c r ib e to w h a t e x te n t th e s e r e s u lt s c a n be c a r r ie d o v e r to a r b it r a r y fre e p r o d u c ts .

§1.

Introduction In a well known paper [14] published in 1 9 3 6 , J . H . C . Whitehead gave

an algorithm to decide whether two elements of a free group of finite rank are equivalent under an automorphism.

An algebraic proof of Whitehead’s

result was later given by P . J . Higgins and R . C . Lyndon [9] and this was the b a s is for three important papers [10, 11, 12] by J . McCool on presen ta­ tions of groups of automorphisms of a free group.

The purpose of the

present a rticle is to outline to what extent th e se results can be carried over to arbitrary free products.

F o r the most part the article is a summary

of [3] and [4], where complete d eta ils are given, but the author has a ls o sought to give a c le a r and e a sily understood description of the e s s e n tia l features of the method, introduced by Whitehead, which has been christened “ P e a k R e d u ctio n .”

Much of what is new or recent in what

follows has been obtained by the author and Heiner Z ie sch an g in the course of a harmonious and stimulating co llaboration.

107

DONALD J . COLLINS

108

We begin with a little notation and terminology.

L et

G = * G- be i |v| ,

(c )

Iu| < jw | or

(u,w,v,cr,r) where

o , r are elements of 12 and

wt = v , |w| > |v| .

Plottin g length upwards, we can v isu aliz e a peak a s one of:

W

V

W

G

We sa y such a peak is reducible (over 12 ) if

07 = Plp2'" PfPi e ^

with lu/ V P j l < H ' j =

1.

109

PEA K REDUCTION

Reducibility can be visualized as w

We say word ( 1 .4 )

12 allow s p ea k red u ctio n if every peak is reducible.

A cyclic

u and p ^, p^, " , p T € 12 form a v a lley if, for 0 < s < t < r , (a)

|up1 - - - p j _ 1 | > |up j - ■p j |,

(b)

|up1 - - - p k_ 1 l = |up1 - - - p k l ,

(c)

l u p j-'-p j jl < lupj — pjl ,

(interpreting e .g .

l < j < s s + 1 < k< t t + 1 < 1< r

s = 0 to mean that (a) is vacuous).

A v alley can be

visualized a s :

The easy part of Whitehead’s argument is contained in the following two propositions.

110

1.5

DONALD J . COLLINS

P R O P O S IT IO N .

S u p p o se

allow s p eak red u ctio n .

c y c lic word u of G and any a e Aut G , there e x is t su ch that y ,

P i > " ’ ’ Pr ^orm a va^ e Y-

Proof.

1) generates

Since

k = l,2 ,* , * ,s .

Aut G , a = o 1

•••crg , with

In general plotting the lengths of

T h en for any p T€ H

e0 , 0 xz

— xz xy - 1 z - 1 y - I

and t : y -> yz .

The way to a ch ie v e peak reduction is to enlarge the generating s e t s o a s to include products su ch a s reducible. ( 2 .1 )

S pecifically

f] is ch o sen to c o n sis t of the automorphisms

and (2 .3 ) the W hitehead autom orphism s, i . e . , those automorphisms

such that, for some fixed

y e S U S ' 1 and any s e S , s o is one of s ,

sy , y- 1 s , y- 1 sy . Clearly that

o r . The above peak is then trivially

Q is finite and Whitehead was able to show

12 allows peak reduction. Now let G = * G- = ( * G-) * F irf id

be an arbitrary free product of

finitely many indecomposable factors

G^. According t o D . I . F o u x e -

Rabinowitsch [6. 7] a generating s et for Aut G can be formed from (2 .4 ) the perm utation autom orphism s, i.e. automorphisms which permute, via fixed isomorphisms, any factors that are isomorphic to one another; ( 2 .5 ) the factor autom orphism s, i . e . , those of form

* y _ 1 Gjy , some

i e J , y k Gj ,

or x -» xy , some

x e S , y 4 x~* .

113

PEA K REDUCTION

The automorphisms ( 2 .6 ) are the analogues of the Nielsen automor­ phisms (2.2 ).

So if the example of an irreducible peak given above is to

be avoided then the generating s e t must be extended to include (2 .7 ) the W hitehead autom orphism s, i .e ., those automorphisms which there is some fixed letter (a)

for any i e J

(b)

for any s e S , s o

(In (a)

either

y (called the operative letter) and y

is one of s , sy , y *s , y 1sy .

o is understood to operate pointwise and if y e

o leav es

a for

, k e J , then

G^ fix e d .)

An u n su ccessfu l attempt to e stab lish peak reduction with this ch oice of £2 is recorded in [2].

The sticking point occurs for peaks involving

two Whitehead automorphisms whose operative letters come from the same factor G^ (not infinite c y c l i c ) and examples of irreducible peaks are given there.

The resolution of the difficulty is achieved by further en­

larging £2 s o that it contains products of Whitehead automorphisms all of whose operative letters lie in the same factor

G^ (not infinite c y c l i c ) .

We refer to th e s e a s m ultiple W hitehead autom orphism s. 2.8 .

PROPOSITION.

L et 0

p le W hitehead autom orphism s.

c o n s is t of a ll perm utation, fa cto r and multi­ T h en fl allow s p ea k red u ctio n .

Proposition 2 . 8 is the principal result of [3].

The proof parallels the

Higgins-Lyndon argument for Whitehead’s original theorem by using a formula which gives the change in length that occurs when a multiple Whitehead automorphism is applied to a c y c l i c word. line of the argument and need the following notation. of exposition, assume no factor is infinite c y c l i c . automorphism with operative letter x e G ^ , write

We give a brief out­ Purely for sim plicity

If o is a Whitehead

114

DONALD J . COLLINS

Then by

o is uniquely defined by the pair

( A ,x ) .

(2 .9 )

(A ,x) and we shall denote

o

It is e a s y to c h e c k that

if A 1 H A 2 = l G ^ i ,

then (A 1 , x ) ( A 2 ,x) = (A1UA2 ,x)

and (2 .1 0 )

(A , x 1 ) ( A , x 2 ) = (A , x 2 x x) , where X j , x 2 e

.

It follows, after a little calcu latio n , that any multiple Whitehead automor­ phism can be written a s a product p = (A1 , x 1) ( A 2 , x 2) ••• (Af ,x r) , where x j e Gk , j = 1 , 2 , — , r , Ap PI Aq =

^ - P' ^ - r anc*’ ^ence

r < |l| .

We c a ll A = A 1UA2 U ••• UAr the domain of p . The reduction of a peak (u,w,v,x 2 ’ ’ ‘ ' ' x r are a ^ a c tual

letters of u . The final ste p , in generalizing Whitehead’s result, is to decide when two c y c l i c words which are of minimal length within their re sp e ctiv e auto­ morphism c l a s s e s are equivalent.

To indicate, albeit very sketch ily , how

this is done, we need some further notation.

Let

£

denote the group of

all permutation automorphisms and the group of all factor automorphisms (se e ( 2 .4 ) and (2 .5 )).

117

PEAK REDUCTION

For c y c lic words u~ v

u and v , which are of minimal length write

if there exists

r e

carrying one of

these representatives into v . Carrying out this last step requires, of course, some unavoidable assumptions about the factors

Gj and their

automorphism groups. When infinite c y c lic factors occur, path components in F longer fin ite.

are no

However, it turns out that the boundedness of the diameter

of a path component s till gives enough control to squeeze out an algorithm. It is difficu lt to say anything very useful about the proof of P rop osi­ tion (2.17) without going into a lot of detail and we content ourselves with two very brief, and only approximately correct, comments. (2.18) At the minimal le v e l, a multiple Whitehead automorphism “ frequently” simply permutes the letters of the c y c lic word on which it operates. any “ lon g” path in V

So

must contain a loop.

(2.19) When no factor of G is infinite c y c lic , an argument somewhat like that for Proposition (2.16) shows that for any minimal c y c lic word

u,

DONALD J . COLLINS

118

there are only finitely many multiple Whitehead automorphisms |u H be the homomorphism induced by inclusion, where H - G * < t 1 , t 2 ,--*,tn > / « w 1 ,w2 ,---,w n » .

Here the group < t 1 ^ 2 ; “ ‘ ,t n >

is freely generated by t 1 ,t2 ,-* - ,tn and the determinant of the matrix of exponent sums of the variables

t^ in the elements

Wj is assumed to be

different from zero (such a map G -> H is called a “ Kervaire e x te n s io n ” ). r

Let gii

z "*>gr r c G and assume that 1 e II i=i gj*i , where gj* i denotes the

co njugacy c l a s s of the image of g^ in H .

Then we conjecture that

r

1 f II gp i=l

(the Kervaire con jectu re is the s p e c ia l c a s e

r = 1 ).

Fin ally we show that F . L e v in ’s argument [6] can be strengthened to show that the reciprocity law is valid for a single equation in one variable of weight

n and exponent sum n (so all occurrences of the variable

occur with positive exponent). I want to thank John Stallings for suggesting that Theorem 1 below may be related to a theorem of Baumslag and Taylor. proved to be the c a s e .

This suggestion

I a ls o want to thank Jim Howie for his careful

reading of the manuscript.

Howie observed the change in focus that

occurs in this paper in §2 from the Kervaire conjecture to the reciprocity law, which was gradually taking place in my mind a s I wrote it.

He su g ­

gested that I inform the reader of this change of emphasis at the start, as I have done here, so the reader will be spared the pains I had to endure in discovering the reciprocity law.

NONSINGULAR EQUATIONS

§1.

123

W eight Let

G be a group and let F

elements

be the free group freely generated by

. An equation over G is simply an element

w e G * F . We may write w uniquely in the form

W = M iJ where

40,

g- e G , si i

and where

••• g/ i T Sr+1

^ tij

4 t-ij+1

if g- = 1 (so no c a n c e lla i r

tion can occur).

We define the weight of w to be

£

i= l

|e-| . Informally,

1

the weight of w is the number of t^ 1 occurring in the reduced e x p r e s ­ sion for w . We sa y that an n-tuple of elements

(h 1 ,h2 ,---,h n) in an overgroup H

of G is a solution of the equation w if w is in the kernel of the com­ posite homomorphism G * F - L

H * F LZ, H ,

where y : F -> H is given by y ( t •) = h- . y\’ The homomorphism G * F >Z sending G to 0 and tj to w2 > "* 'wn € ^

^

c a ^ e(^ nonsingular if

Thus the Kervaire con jectu re s ta t e s that there is a

simultaneous solution of the nonsingular system

w i>w 2 ’ “ *'wn

an over'

group of G . P R O P O S IT IO N

1.

T h e K erv a ire c o n je c tu re is valid if and only if it is

valid for nonsingular s y stem s of equ a tio n s of w eight at most three. The argument is analogous to the familiar procedure of converting a differential equation of high order into a sy stem of first order equations. We shall illu strate the procedure in two exam ples.

Observe that in e ach

c a s e the determinant of the a s s o c ia t e d sy stem of equations of weight < 3 is the same as that of the original sy stem .

124

S. M. GERSTEN

E X A M P L E 1.

w = t 3a t ~ 1b t _1c , where a , b , C f G .

t 2 = t 2a t ~ 1bt~1c , t 3 = tat- 1 bt- 1 c ,

We define

= t ,

= t_ 1 bt_ 1 c , t g = t~ *c . The

a s s o c ia t e d system of equations is

/

W1

=*1*2

\ W2

=*2 •W

' 1

/ w3

=t 3 •( t 1a t 4 )~1

/

w4

=l 4

•( ‘ T 1 b t 5>_ 1

\

W5

=‘ S

'

C)_1

Observe that the original equation w has weight 5 , the number of equations of the a s s o c ia t e d system , and determinant is unchanged. 5-tuple of elements

( h j , h 2 ,h3 ,h4 ,h 5) is a solution of the second system

in an overgroup only if hj E x a m p l e 2.

is a solution of the original equation w . w 1 = x 2aybxc w2 = x

( a ,b ,c ,d ,e

in G ;

The

x ,y

o

dye

indeterminates).

Define

tj = x , t 2 = y ,

t 3 = xaybxc , t 4 = ybxc , tg = x c , t fi = x - 1 dye , and t 7 = ye . The a s s o c ia t e d system of equations is

NONSINGULAR EQUATIONS

125

The sum of the weights of the original system is equal to the number of equations in the a s s o c ia t e d system . seven-tuple

The determinants are equal.

A

(hj ,h2 ,h3 ,h4 ,h 5 ,h6 ,h 7) in an overgroup is a solution of the

a s s o c ia t e d sy stem only if the pair

(h1 ,h2 ) is a solution of the original

sy stem . The argument for the general c a s e , which we omit, shows in fact the following refinement of Proposition 1. P R O P O S IT IO N

L e t w j , w 2 >’ *‘ >wn

2.

a sy stem of equations over G

of p o sitiv e w eigh ts y ^ ,y 2 , “ *, y n re s p e c t iv e ly .

T h en there is an

n

a s s o c ia t e d sy stem of y = 2 y : equations in y 1=1

unknow ns, ea ch of

w eight < 3 , having the sam e determ inant a s the original sy stem and s u c h that (hj ,h2 ,- * - ,h p

is a so lu tio n of the a s s o c ia t e d sy stem in an

overgroup of G only if (h.^,h2 ,**‘ ,hn) is a solution of the original sy stem .

§2.

S y stem s of eq u a tio n s of w eight 2 In considering nonsingular sy stem s of equations of weight at most two,

we may eliminate variables occurring in equations of weight one to obtain a new system with the same determinant and co n sistin g of equations of weight two.

Hence we sh all consider a nonsingular system of equations

w l ' w2 ' * " ' wn over a SrouP G where each wthe indeterminates Let L

K be a

2

in

^ , t 2 ,***,tn .

CW complex with one vertex where ^ ( K ) = G , and let

be obtained from K by a ttaching n 1 -ce lls trivially, so that

771 (KU L*-1

= G * F , where F

the oriented 1 -c e l ls of L - K 77

is of weight ex a c tly

j (K U

is freely generated by

are identified with the elements

), and then a ttaching

C all th ese 2 - c e l l s

a- , l < i < n ,

( ^ us t^ in

n 2 -c e l ls by the maps w- , 1 < i < n . so aj

is attached by word w^ e G * F .

The Kervaire conjecture s ta t e s that the homomorphism ^ ( K ) -> ^ ( L ) induced by inclusion

K U L

is in jective.

From the e x a c t homotopy

S. M. GERSTEN

126

s eq uen ce, we s e e that this is equivalent to showing that the map 772 (L ) -» 7 7 2 ( L ,K )

is su rjectiv e.

a map f [ ( D ^ S 1) -> ( L , K ) .

P-

of L we now describe.

in the interior of the 2 -c e l l a-

the attaching map g ^ S 1 ^ K U L ^

for a j is

weight two, if we fix an interior point Qj L

is represented by

We shall make this map transverse regular to

a tamely imbedded subgraph P Choose one point

An element of 772 (L ,K )

w - , where w-

and

Join

P-'

n U Q - . Thus j=l J

Since has

in the interior of the 1 -c e ll of

corresponding to tj , then we may assum e the map g-

regular to

of L .

is transverse

-l g. ( U Q-) c o n s is ts of precisely two points J

in d a •.

P ^ , P j ' , and

P j by a properly imbedded

PL

interval J j

in a-x

n

and let F

be the image of

U Ji=l

in L .

Since

T

has a regular neigh-

borhood in L , we may make f : ( D ^ S 1) -> (L ,K ) transverse regular to T , and adjust the notation so this is done.

Then f _ 1 ( F )

properly imbedded in D2 . Since f(S1) C K and that A = f - 1 ( r ) that f _ 1 (

is a 1-manifold

R H T = yS, it follows

is a disjoint union of c ir c l e s in the interior of D2 and

U P . ) is a finite union of points subdividing th ese c i r c l e s . l