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A survey of One-Relator Groups Khalifa Alhazaa

Department of Mathematics and Statistics McGill University, Montreal

August, 2005

A thesis submitted to McGill University in partial fulfilment of the requirements of the degree of

Master of Science (MSc) in Mathematics

© Khalifa Alhazaa, 2005

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2

Acknowledgements

l should like to thank Dr. Daniel Wise for his invaluable help and enlightening insight and rereading the MS many times, and Carmen Baldonado, the secretary of the department, for her infinite patience. l should also like to thank Mc Gill University for providing a very suit able environment for education.

Abstract

We review basic concepts of the algebraic theory of combinatorial group theory, and try to emphasize the important role of groups represented with a set of generators and one defining relation.

Dans ce travail, On présente les concepts fondamentaux de la théorie algébrique des groupes combinatoires, tout en montrant le rôle important des groupes représentés par un ensemble de générateurs et une seul relation.

3

CONTENTS

Contents

1 Introduction

2 Background 2.1 Notations and Definitions 2.2 Free Groups . . . . . . . . 2.3 Presentations ...... . 2.4 Tietze and Nielsen Transformations 2.5 Amalgamated Free Products and HNN Extension

3 One-Relator Groups

4 Magnus's Inductive Definition

5 Positive One Relator Groups

6 Small Cancellation Theory

7 The Center of a One-Relator Groups

8 The Tits Alternative

9 One-Relator Groups with Torsion 9.1 Torsion Elements . . . . . 9.2 n-freeness . . . . . . . . . . 9.3 Virtually Torsion-Freeness 9.4 Newman's Spelling Theorem

10 Commutativity of One Relator Groups

Il The Isomorphism Problem

12 Torsion-Free One-Relator Groups

13 Asphericity

14 Exponential Growth

15 One-Relator Product

16 Open Problems

4

5

7 7 7

10 11 14

16

25

34

38

41

45

46 46 49 50 51

56

60

62

67

69

70

73

1 INTRODUCTION 5

1 Introd uction

The study of presentations of groups in terms of generators and relators expanded

during the 20th century to become the field of combinatorial group theory.

While entry to the field do es not require a vast background, the subject is quite

challenging even in the simplest case of a group represented with a single relation. In

fact, the study of one-relator groups has become a substantial subfield.

In this paper we try to give a semi-panoramic view of the subject of one-relator

groups, since there is a unifying theory. Most proofs work by embedding the one

relator group in question in an HNN extension of anothér one-relator group with a

shorter relator, and then appealing to induction.

We start with establishing the background, where we define everything we need

and state all the theorems that will be used later in the paper.

We then define what we mean by a one-relator group. Here we introduce a

rewriting construction that will allow us to embed our particular one-relator group

in an HNN extension of another one-relator group of lower complexity. Weconclude

this section with the proof of the Freiheitssatz, which is a cornerstone of the subject,

and one of the first nontrivial theorems to be proven in this subject. Its proof makes

use of the embedding we mentioned above. The proof we give is a variation of

Magnus's proof. Hruska and Wise used towers in proving the Freiheitssatz, and

similar theorems. Towers, in fact, have the proof of Magnus as their base.

In Section 3, we somewhat extend what we did in Section 2, and that is achieved

by obtaining a finite sequence on which every term is an HNN extension of the

previous term which is of lower complexity. The proof is due to Mihalik, but we

write it in details and rearrange it for the sake of simplification.

In Section 4, we review sorne basic properties of positive one-relator groups and

introduce the important small cancellation theory. Small cancellation theory solves a

variety of the main problems in the theory of one-relator groups. Variants of the small.

1 INTRODUCTION 6

cancellation theory have even been proposed by Juhasz to approach the conjugacy

problem.

In Section 5, we show that one-relator groups are generically small cancellation

in a reasonable sense.

In Section 6, we consider one-relator groups with center. We discuss Murusugi's

result about finding the center of any one-relator group. We then discuss Pietrowski's

solution to the isomorphism problem on one-relator groups with center.

After that, we describe the Tits alternative, and that one-relator groups satisfy

the Tits alternative.

In Section 8, we introduce one-relator groups with torsion and define the notion of

n-freeness that is going to be used in Section 11. We prove that one-relator groups are

virtually torsion-free, and conclude the section with the proof of Newman's spelling

theorem.

In Section 9, we mention results about commutativity in one-relator groups. In

the next section we summarize sorne partial solutions of the isomorphism problem to

one-relator groups.

In Section 11, we introduce two subclasses of one-relator groups, namely, cyclically

pinched and conjugacy pinched one-relator groups. These are obvious generalizations

of surface groups.

In Section 12, asphericity is discussed briefly, and in Section 13 exponential growth

is discussed.

We conclude the paper with Section 14 in which we discuss a direct generalization

of one-relator groups, namely one-relator product. Important theorems of one-relator

groups are extended in two ways: either by restricting the factor groups or the relator

of the one-relator product.

We then list a number of open problems and conclude the thesis with an extensive

bibliography.

2 BACKGROUND 7

2 Background

The material in this section are so general and appear in many references that it is

impossible to give a complete reference. We will direct the reader, however, to [L8]

and [MK8]. In this section we lay the background. We define words, free groups and

presentations. We introduce some indispensable transformations and special products

of groups.

2.1 Notations and Definitions

We denote a subgroup A of G by A :S G, and a normal subgroup N of G by N ~ G.

For a subset X ç G, we define the subgroup of G generated by X by

which is the smallest subgroup of G containing X. And we define the normal clos ure

of X in G by

((X))G = n{NI X ç N ~ G}

which is the smallest normal subgroup of G containing X.

A set X ç G generates G if (X)G = G. 8imilarly X normally generates G if

((X))G = G.

2.2 Free Groups

A group F is free with basis X, if X ç F and for each map <p : X ---* G to a group

G, 3! <p* : F ---* G such that <p*lx = <p.

If Xi is a basis of the free group Fi, i = 1,2, then it is well known (see [L8]) that

2 BACKGROUND 8

Indeed, -{= follows from the universal property and ===? follows by counting homo

morphisms to Z2, i.e. 21xil many.

We define the rank of F to be the cardinality of its base X.

It would be very useful to construct a free group from a basis.

Let X = {Xa,}aEI be a set with sorne indexing set J. We define X-l := {x~l}aEI

to be the set of formal inverses.

A word on X is an expression of the form w = x~: x~: ... x~:, where Xij E X and

Ej = ±1. We denote the set of words on X by W (X), and calI the elements of X±l

letters. If X = {x!, .. . }, we write W (X) as W (Xl,.")

Given any two words w = Xl'" xn and z = YI'" Ym, with Xi, Yj E x±l, i =

1, ... n, j = 1, ... , m, we would like to be able to multiply them. We define their

concatenation as, w . z = Xl ... XnYI ..• Ym' It is obvious that concatenation is an

associative product. We further introduce a special word on X, we calI it the empty

word and denote it bye. The importance of e is revealed in its action as an identity

element with respect to concatenation. In fact, for any word w on X we set w . e :=

w=: e·w.

We remark that W (X) is a free monoid.

An elementary deriving transformation of a word w is obtained by inserting or

deleting a term of the form xx- l or X-lX. A deriving transformation is a successive

repetition of elementary transformations.

If w can be transformed to Wo we say that Wo is derivable from w, and we write

w rv Wo. It is an easy exercise to verify that rv is in fact a free equivalence (congruence)

relation.

If we set X = {[x] 1 X EX}, we see that IXI = IXI.

Claim: The group ft' = W (X)j rv is the free group with basis X.

For let rp : X ---+ H be any map from X to H. If 9 E ft' then 9 = [w], where

w is a word on X. Thus w = X~l ... x~n, and so 9 = [XI]ê1 ••• [xn]ên

• Let rp* (g) =

2 BACKGROUND 9

r.p ([Xl]t 1 ••• r.p ([xn]tn. The mapping r.p* is a homomorphism, sinee if g, h E F, then

9 = [Xlr1 ... [xn]en and h = [Xn+lrn+l ... [xm]em. 80

r.p* (gh) r.p* ([Xlt1 ••• [xmtm

)

r.p ([Xlt1) ••• r.p ([xntn) r.p ([xn+1tn+1

) ••• r.p ([xn]em)

- r.p* ([Xl]éI ... [xntn) r.p* ([xn+1]en+l ... [xntm)

r.p* (g) r.p* (h)

r.p* is unique, sinee if rj; is another homomorphism from F to H, such that rj;lx = r.p

and 9 E F, then 9 = [Xl]e1 ••• [xntn and sinee rj; is a homomorphism, we have

rj;(g) - rj;([Xlt 1) .. ·rj;([xn]en)

- r.p ([XI]t 1 ••• r.p ([xn]tn

- r.p* ([XI]t1 ••• r.p* ([xn]t n

r.p* ([Xlt1 ••• [xnt n

) = r.p* (g).

Thus F is a free group with basis X. This ends the proof of the daim.

But sinee IXI = IXI, then F, the free group with basis X, is isomorphic to F.

Fact 2.1 A subgroup of a free group is free.

For a referenee, the interested reader might consult [MK8] or [Ha].

In the free group F we define the length of w as

{

inf {ni ::lXI, ... , Xn E X±1 with w = Xl'" xn} if w =1= e Iwl =

o if w=e

where e is the empty word.

2 BACKGROUND 10

If x E X, we say that x appears in g, or x appears in the expression of g, or 9

involves x, if :3xil' ... ,Xin E X such that

Let x be a generator. We shall define the mapping ilx : X ~ N as ilx (a) = 1, and

il x (b) = 0 if Y =1= x is a generator.

We may uniquely extend il x to (J x : F ~ N as

where w = x~: ... x~; . G iven an word w we define the exponent sum of q, in w as (J a ( W ).

For instance if w = x 2y-3zx , then Iwl = 7 and (Jx (w) = 3.

The word w = Xl'" X n , Xi E X±l = X II X-l, is reduced if Xi =1= Xiill i =

1, ... ,n - 1, i.e. terms of the form xx-l or X-lx do not appear in the expression

of w. For any given word w, we say that w is cyclically reduced if w2 = WW is

reduced. For instance w = abcd is cyclically reduced, whereas v = abcda-2 is not,

since v2 = abcda-2abcda-2 is not reduced.

2.3 Presentations

Given a group G, we pick a set X with <p : X ~ G such that its extension <p* : F ~ G

from the free group with basis X to G is surjective. Thus G is a homomorphie image

of F. Consequently,

G rv F / ker <f; * . (2.1)

If R normally generates N = ker<f;*, we can rewrite (2.1) as (X)F/ ((R))F, and

2 BACKGROUND 11

denote it by

(X 1 R). (2.2)

This is called a presentation of G, and is by no means unique. Indeed, Ris not a

unique normally generating set of N.

In (2.2) X is called the set of generators of Gand R the set of relators or defining

relations of G. If X is finite, G is finitely generated; if R is finite, then G is finitely

related; and if G is both finitely generated and finitely related then G is finitely

presented. If X = {Xl, X2, ... , xn }, we write G as (XI, X2, ... Xn 1 R) and if R =

{rI, r2,·· ., rn } we write G as (X 1 rI, r2," ., rn ) or (X 1 rI = 1, r2 = 1, ... , rn = 1)

and so on.

2.4 Tietze and Nielsen Transformations

As in Subsection 1.2, an elementary deriving transformation is obtained by inserting

or deleting terms of the form xx-l, x-lX, where x is a generator, or r±l where r is

a relator. Then a deriving transformation is a successive repetition of elementary

derivable transformations. This definition coincides with that of free groups since

free groups have no relators.

The word w is derivable from the relators rI, ... if w can be transformed to the

empty word by inserting or deleting xx-l, x-lX, where x is a generator, or rt l .

Given a presentation (Xl, ... 1 rI, ... ) of the group G, we define an elementary

Tietze transformation to be any of the following:

(Tl) If s is derivable from rI, ... , then add s to the relators of G to get

(T2 ) If the relator ri is derivable from the rest of the relators, then delete ri from

~ ..

2 BACKGROUND 12

the relators of G to get

(T3 ) If w E W (al, .. . ), then add the symbol y to the generators of G and the relation

y = w to the defining relations of G to get

(y, Xl,· .. 1 y = w, rI = 1, ... ) .

(T 4) If ri can be written in the form Xj = v, where v E W (al, ... ) does not involve Xj,

then delete Xj from the generators and ri from the defining relations replacing

it with v, to get

A Tietze transformation is a successive repetition of elementary Tietze transfor

mations.

Tietze showed that applying these four transformations to a presentation does

not change the group. In addition, if G has a particular presentation, then any other

presentation can be obtained from this one by a Tietze transformations.

If r is a relator of C, consider its conjugate h-Irh where h E C, then h-Irh is

derivable from r since we only have to delete r to get h-Ih, which may be transformed

to the empty word by an inductive argument. Thus if the set of relators is R, let R'

2 BACKGROUND

denote R" {r}

G - (X 1 R) = (X 1 r, R')

- (X 1 h-1rh, r, R')

- (X 1 h-lrh, R')

(adding x-lrx by Tl)

(deleting r by T 2)

and so we may replace any relator by any of its conjugates.

13

Now if a relator r is not cyclically reduced, then r = amroa-n such that ro

does not start or end with a±l. Now replace r by it conjugate rI = a-mram =

a-mamroa-nam = roam- n to get rl is cyclically reduced, since ri = roam-nroam-n

which is reduced since ro does not start or end with a±1. Thus we may always replace

any relator by a cyclically reduced relator.

Similar to a Titze transformation, we define an elementary Nielsen transformation

on a system of generators {XI, .' .. ,xn} to be one of the following:

(Nl ) Replace some Xi by xil.

(N2) Replace some Xi by XiXj, where j =f i.

(N3 ) Delete some Xi where Xi = 1.

A Nielsen transformation is a successive repetition of elementary Nielsen trans

formations.

It is easy to show that if a system of generators {Xl,"" xn} of a group G is

transformed to {YI, ... ,Yn} via a Nielsen transformation, then {Yl, ... ,Yn} constitutes

a new system of generators of G.

2 BACKGROUND 14

2.5 Amalgamated Free Products and HNN Extension

Given groups A, Band C and homomorphisms

C ~ A

we define the free product of A and B amalgamating C, A *c B, by the following

universal property:

If

C 'P A -----7

11/1 l

B -----7 G

is commutative, then :3!ç : A *c B -----7 G such that the diagram ~.

C -----7 A

~ ,/

l A*B l c

/' ~ç

B -----7 G

is commutative.

It would be very useful to prescribe a presentation for A *c B from the presenta-

tions of A, Band C.

If A and B have presentations (X 1 R) and (Y 1 S), respectively, where X and Y

are disjoint, it turns out that A *c B is isomorphic to the group presented by

(XIIY 1 RllSll{'P(c) =1j;(c) 1 cE C})

2 BACKGROUND 15

or

(X, Y 1 R, S, <p (c) = 'ljJ (c) Vc E C) (2.3)

In the special case where C C;::! {1} we obtain a free product and denote it by A * B.

We can thus also express the presentation (2.3) of the amalgamated free product

A *c Bas

Now we turn our attention to the Higman-Neumann-Neumann extension, or, for

the sake of brevity, the HNN extension. Given a group G with presentation (X 1 R)

and two isomorphic subgroups K rv H via the isomorphism e, then the HNN exten

sion of G is defined as

G*O rv (G ,tl Clat = e(a) Va E K)

C;::! (X U {t} 1 R, at = e (a) Va E K)

where at = t-1at is a notation for conjugation of a by t.

Theorem 2.2 (Normal Form Theorems) If 9 E G * H, then we can write it

uniquely as 9 = gl'" gn, where gl, . .. ,gn are elements of G or H and whenever

gi E G, then gi+1 E H and vice versa.

If 9 E G *c H, then we can write it uniquely as 9 = cg!'" gn, where c E C,

gl, ... ,gn are coset representatives of G 1 C or HIC and whenever gi E G, then

gi+1 E H and vice versa.

If 9 E G*O:A-->B, then we can. write it uniquely as 9 = gotê1 gl ... tên gn, where

go E G, whenever Ei = -1 then gi is a coset representative of GIA and whenever

Ei = 1 then gi is a coset representative of GI B, and t ê 1aC ê does not appear as a

subword.

3 ONE-RELATOR GROUPS 16

3 One-Relator Groups

A one-relator group is a group with one defining relation:

We begin this section by describing an important rewriting process. We first

illustrate the pro cess with an illustrative example, and after that we describe it in

general in Construction 3.1. We then prove two theorems about embedding a one-

relator group in an HNN extension of another one-relator group with shorter relator.

We conclude the section with the Freiheitssatz.

The groups to which the above mentioned rewriting process is applicable must

have a relator involving at least two generators, one of which must have a zero

exponent sumo

Let G have the one-relator presentation

Notice that (J'x (r) = O. If we consider the second relator to be y, we cyclically

permute r such that it st arts with y. 80 r becomes yxzy2x-3z 3x 2.

If we rewrite every term of the form x-myxm as Ym and x-nzxn as Zn, we may

rewrite r as

Y (xzx-l) (xy2x-1) (x-2z 3x 2)

_ (x-OyxO) (X-(-1)ZX-1) (x-(-1)yx- 1)2 (x-2zx2)3

2 3 1 YOZ-lY-l z2 = r .


Notice that

via the isomorphism <p : Y-l f-+ Yo, Zi f-+ Zi+1'

Let H be the group with the presentation

We observe that Ir'i = 7 and Irl = 11, so Ir'i < Irl. We claim that H*<p, which

has the presentation

is isomorphic to G.

Notice that if i > 1, then Zi = r1Zi_1t = ... - t-izoti, and if i < 0, then

t t -l tt t-1t-1 t 2 t-2 t-i t i Zi = Zi+l = Zi+2 = Zi+2 ... = Zo·

Define 7.jJ* : G -+ H*<p asx f-+ t, Y f-+ Yo, Z f-+ Zo, then

so 7.jJ* is a homomorphism.

so 7.jJ* is also a homomorphism.


In addition, 'ljJ* ('ljJ* (t)) = t, 'ljJ* ('ljJ* (Yi)) = Yi, etc., and we see that 'ljJ* 0 'ljJ* is the

identity. 8imilarly, 'ljJ* 0 'ljJ* is the identity and so G rv H *'P'

Now we consider the general case.

Construction 3.1 Let G = (t, al, ... 1 r) be a one-relator group such that at (r) =

0, we will prove that we may rewrite G in terms of aij = t-jaitj by induction on the

length of r.

If Irl = 1 then at (r) = 0 can only hold if r = a, where a =1- t is a generator. 80

rewrite r as r' = ao.

We assume now that for any G as above with Irl < m, the word r may be

rewritten in terms of the aij 's, and let H be any finitely generated one-relator group

with relator s such that Isl = m.

If t does not appear within r = a:1 ••• a:n rewrite r as r' = a:1

••• aén• If, on

tl tn ' t10 tno

the other hand, t appears somewhere in s, we may assume that t±l appears in the

beginning of s, or else cyclically permute s so that it starts with t±l. 80 3p, q E Z and

k E {l, ... ,n} q =1- 0 such that s = tPa%so, where So is a word on t, al ... ,an which

does not start with a,/:l. In particular s = tPa%rptpso.

Notice that

0= at (s) - at (tPa%CPtPso)

- at (tPa%CP) + adtPso)

O+at(tPso)

so aa (iPso) = 0, and liPsol < liPa%sol = Isl, so by induction iPso can be rewritten in

term of the aij 's to give s~. Thus


which provides a rewriting of s in terms of the aij 's.

We are now ready to prove our first theorem about one-relator groups, namely,

Moldavanski'l's theorem. The pro of can be found in [LS].

Theorem 3.2 IfG has the presentation (t, a, bl , ... 1 r), with at least two generators,

where r is cyclically reduced and t occurs in r with (Jt (r) = 0, then there is a one

relator group H whose relator is shorter than r, such that G is an HNN extension of

H. lndeed, G rv H *'1' as in (3.1) below.

Proof. We arrange generators such that t, a, appear in r where (Jt (r) = 0 and r

begins with a. Set aj = t-jatk , bij = rjbitk and, by Construction 3.1, rewrite r as a

product r' of aj 's and bij 's so ao appears in r'.

Since t appears at least once in rand never in r' and the aij 's correspond to the

ai's, then Ir'i < Irl· Then set

m min {jl aj appears in r'}

M - max {j 1 aj appears in r'}

And let H be the group with presentation

Now set

K - (am, ... ,aM_I,blj,b2j,'" ;j EZ)H'

K' (am+1, ... ,aM,blj,b2j,'" ;jEZ)H'

Notice that K, K' < H, and that cP K ----7 K' defined by aj I-t aHI, j


m, ... , M - 1 and bij ~ biH1 , i, j E il is an isomorphism of groups. Consequently,

And as in the illustrative example bij = Cibiot and aj = t-ibot.

Now eonsider 'ljJ* : G -+ H*'P sueh that x ~ t, Y ~ Yo, Z ~ Zo and 'ljJ* : H*'P -+ G

sueh that t ~ x, Yi ~ x-iyxi, Zi ~ x- i zx i , we see, as in the example, that 'ljJ* and 'ljJ*

are homomorphisms and that 'ljJ:;1 = 'ljJ*.

Thus we have established that G r...; H *'P' •

The following theorem is used many times in the se quel. The proof ean be found

in [L8].

Theorem 3.3 Given a one-relator group G with a nonempty relator rand at least

two generators, we can embed G in an HNN extension of a group whose relator is

shorter than r.

Proof. If r involves a generator a sueh that rJa (r) = 0, then by Theorem 3.2

(page 19) we can find a group H with a relator shorter than that of G, such that G

is an HNN extension of H. Take the identity to be the desired embedding.

If, on the other hand, r does not involve any element with zero exponent sum,

then either r involves one generator, in which ease there must be a nonzero integer

n so that r = an, or r involves at least two generators.

In the former case, i. e. r = an for sorne non zero integer n. If t is another generator

of G, then rJt (ro) = 0, and we may apply the first case.

In the latter case, i. e. r involves at lease two generators in its expression, say a

and b, we introduee the map 'ljJ : G -+ C given by a ~ yx-(3, b ~ xa and Ci ~ Ci for

any other generator, where G = (a, b, Cl, ... 1 r), and


where a, b, CI, ... , Cn, are the letters appearing in r. Set a := aa (r) and {3 := ab (r).

Now cyclically reduce r (yx- f3 , xQ, Cl, ... , Cn) to rI = rI (y, x, Cl, ... , Cn), then y

occurs in rI and ax (rI) = ax (r (yx- f3 , x Q, a3"")) = -{3aa (r) + aab (r) = -{3a +

a{3 = O. Thus by the first case we may write C as an HNN extension of a group

whose relator s is a rewriting of rI, which implies that the x's appearing in rI will be

omitted and thus Ir'i < Irl.

m m m

L ab (rni) = L niab (r) = ab (r) L ni i=l i=l i=l

m m

i=l i=l

but aa (r) =1 0, and thus 2:::::1 ni = 0, which in turns means that k = 0 or that b has

infinite order in G, i.e. (b)G is freely generated by b.

Now

Consider (x 1 - ) and its subgroup (xQ). Since b has infinite order then b I-t xQ

establishes an isomorphism (b) G ---+ (xQ), and noticing that

C' 0:: (a,b,cl, ... lr(a,b,cl,""cn)) * (xl-) (b)G~(xa)

0:: G * (x 1 - ) (b)G~(xa)


then 'ljJ : G --+ C is an embedding as desired. _

It is customary to prove theorems of one-relator groups by induction on Irl. In this

case we usually prove the assertion first in the case wh en r involves a single generator,

instead of the initial induction step, sinee the latter follows from the former.

One-relator groups play a major role in combinatorial group theory, being the

next simplest ex amples of finitely presented groups. One is inclined to regard this

type of groups as being easily characterized. The truth, however, is that the theory

behind this class of groups is difficult enough.

Theorem 3.4 (Freiheitssatz) Let G have the presentation (al, ... 1 r), where r is

cyclically reduced. If L c {al, ... } omits one generator appearing in r, then (L) G is

freely generated by L.

This theorem was discovered by Magnus who proved it in his Ph.D. dissertation,

which opened the door to the one-relator group arena. It was announced in [Ne68].

The proof we are going to supply below can be found in [LS].

Any subgroup satisfying the hypothesis of the Freiheitssatz, i. e. omits one gener

ator appearing in r, will be called a Magnus subgroup.

Pro of. We shall consider sets that contain aIl the generators exeept for one

involved in r. For otherwise, if L is any other subset of the generators omitting a

generator appearing in r, say al, then L c {a2, ... } which contains aIl generators but

one involved in r, and if the (a2," ')G is freely generated then so is the (L)G' sinee

the subgroups of any free group are free, as in Fact 2.l.

We proceed by induction on Irl. We first prove the case when r involves one

generator, say a. In this case there is an integer n such that r = an. If we rewrite

the presentation of G as


and since L can only omit a, then L = {b1 , ... } and thus (L)Q = (bl, .. ')Q which is

free.

So for the general case we may assume that G has at least two generators a and

t. And we do that in two cases:

Case 1: Sorne generator appearing in r, say t, has zero exponent sumo

There are two possible cases; either L omits t or it omits another generator, say a.

We rewrite G as (t, a, b1 , ... 1 r), where t, a, b1, ... , bk are all the generators appearing

in r. By Theorem 3.2 (page 19) G can be exhibited as the HNN extension of the

one-relator group H with the presentation

where Ir'i < Irl. Thus by induction it follows that

are both freely generated in H.

If L = {a, b1 , ... }, then at least one generator of H wi th nonzero su bscri pt occurs

in r'. Identifying each a, bi i = 1, ... , with ao, bio , respectively, we get that (L) H :S K

or K', and is thus free.

If, on the other hand, L = {t, b1 , ... }, let w be a reduced word on L. If W =Q 1,

So if at (w) =F 0, then w =Fa 1, and thus L freely generate a free subgroup.

If, on the other hand, at (w) = 0, rewrite w as a word on J = {b1i , b2i , ... (i E Z)}


to obtain the reduced word w* and that (J) B is freely generated by induction. So

w* =lB 1. But since w* =a W then w =la 1 and thus (L)c is freely generated.

Case 2: No generator involved in r has zero exponent sumo

Rewrite the presentation of G as (a, b, Cl, ... 1 r) and Since r involves the gener

ators a, b, Cl, ... , Cn , and L = {b, Cl, ... }. By Theorem 3.3 (page 20) we may embed

Gin

c = (y, X, CI, ... 1 rI (y, x, Cl,"" Cn ))

via the homomorphism a f--+ yx- f3 , b f--+ xŒ and Ci f--+ Ci, i = 1, ... and C is an HNN

extension of a group whose relator is shorter than r.

By induction (x, Cl, .. ')c is freely generated, but since (xŒ, cl,·· ')c '" (i, Cl," ·)c'

then (XŒ, Cl, .. . )c is freely generated. Yet, again, 'I/J : (L)c - (xŒ, Cl,·· .)c' and thus

(L) G is freely generated. _

From this remarkable theorem we get a number of applications. For example,

every finitely generated one-relator group can be embedded as a subgroup of a two

generator one-relator group. In fact every countable group with n relators can be

embedded in a two generator n-relator group, a fact due to Neumann [N54].

(~

4 MAGNUS'S INDUCTIVE DEFINITION 25

4 Magnus's Inductive Definition

All the results in this section are based upon [Mi92].

Before we state the inductive definition, we digress into the rewriting of a finitely

generated one-relator group.

If O"a (r) =1- 0 for any generator a, we still can do sorne rewriting, but this time a

rewriting of G * Z. We willlimit ourselves to the case when r involves at least two

generators for practical reasons.

We shall start with an illustrating example as we did in Section 2.

Let G have the presentation (a, b, c 1 a2b3 c2) , and notice that none of the gen

erators has a zero exponent sum in r = a2b3c2• As a result, G * Z will have the

presentation (a, b, c, x 1 a2b3c2).

We set Ct = 0" a (r) = 2 and (J = O"b (r) = 3, and adding y, z to the generators

and y = ax3 and z = bx-2 to the relators of G * Z we get a new presentation

of G * Z as (a,b,c,x,y,z 1 a2b3c2,y = ax3 ,z = bx-2), since we are only applying a

Tietze transform.

But by the same argument

rv ( b 1 2b3 2 - -3 b - 2) = a, ,c,x,y,z a C ,a-yx , -zx

rv / b 1 ( -3)2 ( 2)3 2 - -3 b _ 2) = \a, ,c,x,y,z yx zx c ,a - yx , - zx

O"x (rI) = O. So we are back to the setting of Theorem 3.2 (page 19), and we may


rewrite rI as

and H is given by the presentation

Notice now that Iril = 7 = Irl and that ri involves six different generators whereas

r involves only three.

Our problem now is to inspect when ri has more generators than rand when not.

Now in the general case, let x be a formalletter different from the generators of G

and let G * 7l have the presentation (a, b, Cl, ... ,Cn , x 1 r) with a, b, Cl, ... ,Ck being an

the generators appearing in r, 0 ::; k ::; n. For the rest of this chapter set 0: := (la (r),

and (3 := (lb (r).

Add to the generators of G * 7l the two formalletters y and z, and to the relators

of G * FI the two relators y = ax f3 and z = bx-a . This does not change the group,

because it is a mere application of a Tietze transform. Solving for a and b in terms

of y and z, respectively, we obtain a = yx- f3 and b = zxa , and substituting these two

into the expression of r (a, b, Cl, ... ,Ck) we get the word rI (Cl, ... ,Cn , x, y, z). Now

eliminate the redundant a, a = yx- f3 , b, b = zxa , which would give a new presentation

for the group G * 7l because it is a Tietze transformation.

This leaves us with the presentation


Now since a = yx-/3 and b = zxO<, then

(4.1)

and we are back to the previous case. So we rewrite Tl as T~ in terms of Cij 's, x/s"

Yj 's, and Zj 's. Let H have the presentation

(4.2)

Notice that whenever a occurs in T we have the same number of occurrences of Yj 's

in T~. The same goes for b with z/s, and Ci with Ci/S, so we conclude that IT~ 1 = ITI. Now we inspect the possibility of T~ having more generators of Ci/S, x/s, y/s,

and z/s than T of Cl, ... ,Cn , x, y, and z. The following lemma gives the desired

characterization.

Lemma 4.1 If G = (al, ... ,an 1 T (al, ... ,ak)) then T~ as given above involves mOTe

genemtors than T unless G has the presentation

Proof. Each occurrence of ai in r corresponds to a triple (s, u, t), where u = a;l

and r = sut.

For instance, if r = abc2 b-2 a3 cb, then the first occurrence of b corresponds to

(a, b, c2b-2a3 cb) and the second occurrence to (abc2 , b-I, b- I a3cb).


Now for any letters a, band u and words s and t, define

aa (s) ab (sut) - ab (s) aa (sut) if u i- a-l, b- l

Ia,b (s, u, t) = (aa (S) - 1) ab (sut) - ab (s) aa (sut) if u = a-l

aa (s) ab (sut) - (ab (s) - 1) aa (sut) if u = b- l

If, for example, r = sut and u = c±\ then aa (sut) = aa (r) = Œ, ab (sut) -

ab (r) = f3 and

Ia,b (s, u, t) = f3 (aa (s) - 1) - Œab (s)

f3aa (s) - Œ (ab (s) - 1)

-ax (S2) if u = a-1

-ax (S3) if u = b- l

where S1, S2, S3 are the initial segments of rI before the corresponding occurrence

of u, y-l, z-l, respectively, in rI. The truth of this can be seen manifestly with an

argument similar to (4.1).

80 this occurrence of a;l in r becomes

ain if i- -1 b-1 ai a ,

Yn if ai = a-l

Zn if ai = b- l

where n = Ia,b (s, u, t).

Thus we get more generators in r~ than in r if for sorne occurrence of a letter ai

in r = sut = s'u't', u = a;l = u' and Ia,b (s, u, t) i- Ia,b (s', u', t').

To simplify we restrict ourselves with the case aai (r) > 0, i = 1, ... , k, and the

other case is similar. We consider the following exhaustive four cases:

Case 1: a, b appear in r with aa (r) > ab (r)

This is plausible since r involves at least two distinct generators by our assump-

tion. If necessary cyclically permute r such that r st arts with ai and Partition it

4 MAGNUS'S INDUCTIVE DEFINITION

as

where a = (J'a (r) and (J'a (Si) = 0 Vi = 1, ... , a.

80 by assumption

a

Œ = (J'a (r) > f3 - (J'b (r) = (J'b (aSlaS2··· aSa) = L (J'b (aSi) i=l

a a

- L ((J'da) + (J'b (Si)) = L (J'b (Si) i=l i=l

29

and so minl:Si:sa {(J'b (Si)) < 1, but (J'b (Si) E NU{O} Vi = 1, ... ,a, so minl:Si:sa {(J'b (Si)) ~

O. Renee 3j E {l, ... , a} such that (J'b (Sj) ~ O. We may cyclically permute r again

so that this Sj coincides with SI. 80 without 10ss of generality (J'b (SI) ~ O.

Now r = (1) (a) (SlaS2 ... aSa) = (aSl) (a) (S2· .. aSa), whereas

Ia,b (1, a, SlaS2··· aSa) - (J'a (1) (J'b (r) - (J'b (1) (J'a (r) = 0

Ia,b (asl, a, S2··· aSa) - (J'a (aSl) (J'b (r) - (J'b (aSl) (J'a (r)

Case 2: Vi = 1, ... ,k (J'ai (r) = m =1- 0 and 3 a 1etter c such that both c and c-1

appear in r.


In this case if r = sut

aa (s) ab (r) - ab (s) aa (r) if u =1 a-I, b-1

Ia,b(S,u,t) - (aa (s) - 1) ab (r) - ab (s) aa (r) if u = a-1

aa (s) ab (r) - (ab (s) - 1) aa (r) if u = b-1

m (a a (s) - ad S ) ) if =1 -1 b-1 ua,

m (a a (s) - ad s) - 1) if u = a-1 (4.3)

m (a a (s) - ab (s) + 1) if u = b-1

length over aU pairs of occurrences of letters and its inverses. Say, without loss of

generality, that r = SOaSI a-1 S2 and let b be any let ter appearing in SI.

Now, since neither of a nor a-1 occur in SI (otherwise contradicting the minimality

of SI), then aa (SI) = 0, and again appealing to the minimality of SI we see that

both band b-1 cannot occur in SI, thus ab (SI) =1 O. Thus r = (SOaSl) (a- 1) (S2) =

(Sa) (a) (Sla-ls2)

Ia,b (soasI, a-1, S2) - m (aa (SOaSl) - ab (SOaSl) - 1)

- m(aa (sa) + aa (a) + aa (SI) - ab (Sa) + ab (a)

+ab (SI) - 1)

- m (aa (sa) + 1 - ab (Sa) + ab (SI) - 1)

- m (aa (sa) - ab (Sa) + ab (SI))

Ia,b (Sa, a, Sla-ls2) - m (aa (Sa) - ab (Sa))

so


since (Jb (SI) =1= O.

Case 3: Vi = 1, ... , k (Jai (r) = m =1= 0, every let ter appears with only positive

exponents, and there appears in r two letters a and b such that b does not occur or

it occurs more than once between pairs of successive occurrences of a.

In this case, each occurrence of a would correspond to a different ai in the rewriting

of r, and we would have more generators in r' than in r.

Case 4: Vi = 1, ... ,k (Jai (r) = m =1= 0, every let ter appears with only positive

exponents, and V pair of letters a and b appearing in r, b occurs between pairs of

successive occurrences of a exactly once.

lailaj ((al' .. ak)l al ... ai-l, ai, ai+l ... ak (al' .. ak)m-l-l)

- m ((Ja, ((al" . ak)l al" . ai- l ) - (JaJ

(al'" ak)l al" . ai- l ) )

_ {lm - (l + 1) m if j::; i - 1 -m if j::; i - 1

lm - lm if j > i 0 if j > i

That is to say, for any possible partition sut of the relator (al'" akt, we have

that la. a is not altered. 80 the number of generators in r' coincides with the number " J

of generators in r.

Thus, by inspection, the only case in which the number of generators is equal in

r' and r is Case 4 as desired. _

Now we are ready to state and prove Magnus's inductive definition.

Theorem 4.2 Let G be a finitely generated one-relator group, then there is a finite

sequence of finitely generated one-relator groups, Hl, H2, ... ,Hn = C, such that:

2. Vi < n, either Hi+l or Hi+l * Z is an HNN extension of Hi'


The reason this theorem is named after Magnus is because its proof is patterned

after the proof of the Freiheitssatz.

Proof. We will proceed by induction on n = Irl, where G = (al, a2,"" ak 1 r). If n = 0, G is free and so the desired sequence is {G}.

If n > 0, we have the two following cases.

Case 1: (n - k = 0)

This implies that aIl generators of G appear in r with power 1. So, by a possible

rearrangement of the generator

where the last step is attained by applying a Tietze transformation.

So G is free .and thus the sequence is again {G}.

Case 2: (n - k > 0)

It suffices to find a one-relator group H whose presentation is

with n' = Ir'i < n or n' = n and n' - k' < n - k, and such that either G or

G * Z is an HNN extension of H. In which case we may apply induction on H to

obtain a sequence {KI, K 2 , ..• , Kt} satisfying Magnus's inductive definition. Hence,

setting KHI := G, we get {KI, K 2 , ••• , K t+1} as a sequence for G satisfying Magnus's

inductive definition, completing the proof.

So to find such an H, we need to regard the following cases:

Case 1: r involves a generator a such that aa (r) = O.

This is granted by Theorem 3.2 (page 19).


Case 2: 'ïI generator a appearing in r such that (J'a (r) =1=- O.

From the above discussion and Lemma 4.1 (page 27), we have that either r~, r

involve the same number of generators or r~ involves more generators than r.

In the former case and by Lemma 4.1 (page 27) we have

In the latter case, namely r~ involves more generators than r, set ri = r~ in (4.2)

to get ni = n and ni - k' < n - k since k' > k. •

5 POSITIVE ONE RELATOR GROUPS 34

5 Positive One Relator Groups

Definition 5.1 A word w in a;l's is positive if w is a word in ail, i.e. no inverses

appearinw.

Definition 5.2 A presentation (al, ... 1 rn ) is positive if ris a positive word, n 2:: l.

We calI C a positive one-relator group if it has a positive presentation (al, ... 1 rn ).

If H, K ~ C are subgroups, then [H, K] = {rh, k] 1 h E H, k E K}, where rh, k] :=

hkh-Ik-l is the commutator of h and k.

The ith derived group of a group C, denoted by C(i), is defined by

C(O) _ C

C(iH) _ [C(i) , C(i)]

Definition 5.3 A group C is solvable if::ln EN" {a} such that c(n) = (1). We will

let S denote the collection of aIl solvable groups.

N ow if we define

Cl - C

Ci+l - [Ci, Cl

then the sequence {CJ:I is called the lower central series of C.

Definition 5.4 A group C is nilpotent if::ln EN" {a} such that Cn = (1). We will

let N denote the collection of all nilpotent groups.

Definition 5.5 If X is a class of groups (e.g. Sor N), then a group C is residually

X if given any nontrivial g E C we can find N ~ C S.t. g t/:. N and CI N E X.


The group G is poly X if there is a normal sequence

such that Gi+1/ Gi E X.

The group G is poly-residually X if G is poly RX where

RX = {G 1 G is residually X} .

We remark that not every group is residually solvable. For instance,

G = (a, b 1 a = [a, b-IabJ) (5.1)

is not residually solvable. lndeed, a E G(n) for each n.

On the other hand, Baumslag [Bau71] proves that every positive one-relator group

is residually solvable. Moreover, he proves

Theorem 5.6 Every positive one-relator group has a finite invariant series

such that:

1. GI is free.

2. If n = max {Igl : g E G, Igl < oo}, then G2 / GI = KI EB ... EB Kz EB H,where

Ki are cyclic groups such that IKil = n and H is a free abelian group.

3. Gi+d Gi is residually torsion-free nilpotent.

In addition, Magnus [Ma35] proved that free groups are residually nilpotent, and

the proof can be found in English in [MKS]. This along with the above theorem


imply that every positive one-relator group is poly-residually nilpotent. Furthermore,

every torsion-free positive one-relator group G is poly-residually torsion-free nilpotent.

We may also deduce that any positive one-relator group is poly-residually finite. It

is, however, not true that every one-relator group is poly-residually finite. Take for

instance the nonpositive one-relator group given in (5.1). See [Bau71].

Definition 5.7 Let G with the presentation (al, .. ' 1 rn) be positive, a piece in G

is a word that appears in two different ways as a subword of the cyclic word r. We

denote the set of pieces of G by P ( G) .

Definition 5.8 The positive one-relator group G ~ (al, ... 1 rn) satisfies C' (Ct) if

Wise [Wi01] proves the following theorem.

Theorem 5.9 Let G be a positive one-relator group satisfying C' (1/6)) then G is

residually finite.

Now we introduce the following set of notations:

B (n, m) - {G = (al, ... , an 1 r) 1 ris positive S.t. Irl = m},

Q (n,m) - {GEB(n,m) 1 G do es not satisfy C' (1/6)} , m

13 (n,m) UB(n,k), k=l m

Q (n, m) UQ(n,k). k=l

In the same paper, Wise states that positive one-relator groups are generically

C' (Ct) for any Ct in the sense that

IQ(n,m)1 ~ 0 lB (n, m)1 m->oo


which means, in conjunction with Theorem 5.9) that positive one-relator groups are

generically residually finite.

It is not true, however, that any positive one-relator group is residually finite. For

instance take the one-relator group given in (5.1). we see that

(a, t 1 r l a2t = a-3) ~ (a, t 1 r l a2ta3)

~ (a, t, b 1 r l a2ta3, a = tb)

~ (a, t, b 1 rI (tb)2 t (tb)3 ,a = tb)

~ (a, t, b 1 btbt (tb)3 ,a = tb)

~ (a, t, b 1 btbt (tb)3 ,a = tb)

~ (t,blbtbt(tb)3)

and we see that it is a positive one-relator group but not residually finite.

6 SMALL CANCELLATION THEORY 38

6 Small Cancellation Theory

As suggested by Section 4, there are many interesting connections between one-relator

groups and small cancellation theory. One such possible connection is posed by

Juhasz [Ju91, Ju92] who devised a small cancellation theory in an attempt to attack

the conjugacy problem. The Newman spelling theorem, discussed in Section 8, is

another connection between small cancellation theory and one-relator groups with

torsion. More results can be found in [KSc05].

We remark that Pride [Pr83] proved that one-relator groups satisfy C (2n) (defined

below). The proof is not elementary.

A one-relator group with torsion having the presentation (al, ... 1 rn ) is C (n) if

r =J Pl' .. Pm with m < n and Pi are pieces.

In this section we are going to prove that not only positive one-relator groups are

generically residually finite, but also one-relator groups.

Following the same language of Section 5, we define

o (n,m)

S (n, m)

O(n,m)

S (n, m)

{G = (al,"" an 1 r) 1 Irl = m},

{G E 0 (n,m) 1 G does not satisfy C' (an m

UO(n,k) k=l m

US(n,k) k=l

In P (n, m) we shall not distinguish between isomorphic presentations, and will iden-

tif Y each such presentation with the word corresponding to its relator. Now we will

prove that if 0 < a < 1 then

IS(n,m)1 -> 0 IO(n,m)1 n--oo

The proof of this result follows that of Wise [WiOl].

Proof. The number of words of length m on n letters is (2n)m, counting the


inverses. Consider the action of Zm on this set by ail ... ain 1--+ ai2 ... ain ail whose

orbits have at most n elements.

Nowa word wk E 0 (n, m) has m = Iwl elements in its orbit if w is not a proper

power, and thus 10 (n, m)1 ;:::: (2n)m / m.

On the other hand, if w E 8 (n, m), then 3P, a piece, appearing in two different

ways in w, but Ipl ;:::: a Iwl. We may cyclically permute w to ensure that w st arts

with the subword P (noting that cyclically permuting the relator of G results in two

isomorphic presentations which are the same according to our convention).

There are two possible cases,

Case 1: The two pie ces do not overlap, in which case w = P8PT for some words

8 and T.

The number of positions of P is at most (2n)lsPTI = (2n)lwl- IPI.

Case 2: The two pieces do overlap, in which case w = ABC D with P = AB =

BC.

Notice that ABC is completely determined by A. Thus the number of possible

positions of P is less than (2n)IAI+IDI = (2n)lw l- IBCI = (2n)lwl- lPl .

In either case, there are at most (2n)lwl- IPI possibilities for positions of P and m =

Iwl ways ofpositioning the second occurrence of P, so 18 (n, m)1 :::; m (2n)lw1- 1PI. How

ever, Iwl-IPI :::; Iwl-a Iwl = (1 - a) m, which means that 18 (n, m)1 :::; m (2n)(1-a)m.

80 at length,

lB (n, m)1 10 (n,m)1

m m L: 18 (n, m)1 L: k (2n)(1-a)k

_k=_l ____ < k=l

f: (2~)k m

L: 10 (n, m)1 k=l k=l

But ~m (2n)k > (2n)m/ m and ~m k (2n)(1-a)k < ~m m (2n)(1-a)m = m (2n)(1-a)m ~m L-k=l k - L-k=l - L-k=l L-k=


m2 (2n)(I-a)m, so finally

18 (n, m)1 < m2 (2n)(I-a)m _ m3

10 (n, m)1 - (2n)m (2ntm ~ 0, m

as desired.

In addition, given any piece in G = (al,' .. 1 rn ), then P, as defined ab ove , is a

subword of the cyclically reduced word r that appear twice, so

and thus G satisfies C' ( 1/ n) •

7 THE CENTER OF A ONE-RELATOR GROUPS 41

7 The Center of a One-Relator Groups

In this section we are going to state Murasugi theorem about the center of one-relator

groups and Pietrowski's solution to the isomorphism problem of one-relator groups

with center.

In [MKS] it is proven that if 9 E A * Band both a, gag-1 E A*, then 9 E A.

But since A * B = B * A, then the same argument applied to this yields the same

conclusion about B. We claim now that the centre of the free product is trivial.

Proof. If x E Z(A * B), then \:jc E A * B, cx = xc. In particular, if a E A we

have ax = xa or xax-1 = a E A and thus x E A by the above. Similarly x E B, and

thus x E A n B = (1). •

M urasugi [M u64] stated and proved the following stunning

Theorem 7.1 If G is an one-relator group, then:

1. If G is generated by at least three generators, then Z (G) = {1}.

2. If G is abelian an generated by two generators, then Z(G) = G.

3. If G is nonabelian an generated by two generators, then Z (G) = {1} or Z (G) rv

/Z.

Baumslaug and Taylor devised an algorithm to compute the center of the one

relator group G.

We next turn our attention to the isomorphism problem for one-relator groups

with a nontrivial center.

Definition 7.2 Let {AJiEI be a collection of groups and for sorne pairs of groups

Ai and Aj there is an isomorphism (}ij : Uij (~ Ai) ----+ Uji (~ Aj), such that eji = e;/. The partial generalized free product of Ai 's is


To each Ai we assign a vertex Vi and if there is ()ij : Uij (:S Ai) ---7 Uji (:S Aj) we

join Vi to Vj by an edge Eij. If this graph is a tree we call G a tree product with

factors Ai amalgamating Uij and Uji under ()ij and denote it by

If any subtree of the above tree has at most two extremal vertices, we say that G

is a stem product.

Pietrowski [Pi74] proved that any noncyclic group with nontrivial center is either

the tree product of cyclic groups

(7.1)

where Pi, qi 2': 2 and (Pi, qj) = 1; .or of the groups

(7.2)

where Pi, qi 2': 2, TI:~1 Pi = TI:~1 qi and (Pi, qj) = 1 for i > j.

AIl the theorems below in this section can be found in [Pi 74].

Theorem 7.3 Let G be a noncyclic one-relator group with a nontrivial center sup

pose G/ [G, G] is not free abelian of rank 2, then G has presentation (7.1).

Theorem 7.4 Let G have the two presentations (7.1) and

lb b 1 brl - bSl brn

-l

- bSn - l \ \ 1,···, n 1 - 2"'" n-l - n 1

Theorem 7.5 Let G be a one-relator group with a nontrivial center, su ch that G/ [G, G]


is a free abelian group of rank 2, then G has presentation (7.2).

Theorem 7.6 Let G have the two presentations (7.2) and

It b b 1 br1 - b81 brn

-1

- bSn-

1> \ ' 1,···, n 1 - 2"'" n-l - n

some 1 ::; i ::; n - 1

{ ri+j-l if l::;j::;n-i Pj -

ri+j-n if n-i::;j::;n-l

{ Si+j-l if l::;j::;n-i % -

Si+j-n if n-i::;j::;n-l

or

{ Si+j-l if l::;j::;n-i Pj -

Si+j-n if n-i::;j::;n-l

{ ri+j-l if l::;j::;n-i qj -

ri+j-n if n-i::;j::;n-l

Lemma 7.7 If G has the presentation (7.2) then G has the presentation

It b b 1 br1 - bS1 brm

-1

- bSm-

1 > \ ' 1,···, n 1 - 2"'" n-l - n

r--,


where for some 1 ~ i ~ m - 1

{ Pi+j-l if 1~j~n-i

rj -Pi+j-n if n-i~j~n-1

{ qi+j-l if 1~j~n-i Sj -

qi+j-n if n-i~j~n-1

or

{ qi+j-l if l~j~n-i rj -

qi+j-n if n-i~j~n-1

{ Pi+j-l if l~j~n-i

Sj -Pi+j-n if n-i~j~n-l

Let Gand H be two finitely generated one-relator groups with nontrivial centers.

Since aU finite presentations of Gand H can be recursively enumerated by a repeated

application of Tietze transforms, we can devise an algorithm to find a presentation

of the form (7.1) or (7.2) for each of Gand H. Then by inspection and the use of

Theorems 7.3-7.6 (pp. 42, 43) and Lemma 7.7 (page 43) we can see whether Gand

H are isomorphic or not.

8 THE TITS ALTERNATIVE 45

8 The Tits Alternative

A collection of groups C satisfies the Tit's alternative if VG E C 3H ~ G so that H

is nonabelian free or 3K ~ G so that K is solvable and [G : K] < 00.

One-relator groups were proved to satisfy the Tit's alternative by Karrass and

Solitar [KS71]. In fact Karrass and Solitar proved the following finer theorem.

Theorem 8.1 Every subgroup of a one-relator group either contains P2 or is solvable.

The proof will depend upon the following Lemma.

Lemma 8.2 If G is the HNN extension of a group K, such that 3s E N such that

VH ~ K then H is either solvable of length ~ s or P2 ~ K, then G is either solvable

of length ~ s + 2 or P2 ~ G.

Proof of Theorem 8.1. If G has the presentation (x 1 xn), then G is abelian

and thus is solvable.

So we consider one-relator groups with at least two generators. We will proceed

by induction on Irl, where G has the presentation (al, ... 1 r).

If Irl = 1, then r = ai for sorne i, say r = al without loss of generality, thus G

has the presentation

and so Gis free, so it contains a free subgroup of any rank.

Now assume that one-relator groups with Irl = n satisfy the Tit's alternative, and

let G be a one-relator group with relator r such that Irl = n + 1. By Theorem 3.3

(page 20) we may embed G in an HNN extension of a one-relator group K with

relator s such that Isl ~ n. By induction every subgroup of K either contains P2 or

is solvable of length n, and thus by Lemma 8.2 every subgroup of G' either contains

P2 or is solvable of length 2n. In particular every subgroup of G with finite index is

solvable of length 2n or contains P2 . •

9 ONE-RELATOR GROUPS WITH TORSION 46

9 One-Relator Groups with Torsion

In this section we define torsion elements and state sorne important theorems char

acterizing torsion elements in one-relator groups. We also introduce the notion of

n-free one-relator groups, which will be helpful in Section 11, and show that one

relator groups are virtually torsion-free. We th en conclude the section with Newman's

spelling theorem, which plays a major role in the connection between one-relator

groups and small cancellation theory.

9.1 Torsion Elements

A nontrivial element 9 E G has order nif (g)c rv Zn' infinite arder if (g)c rv Z, and

the identity element will be the only element with order zero. We den ote the order

of 9 by 0 (g).

The nontrivial element 9 is a torsion element if there is a natural number n 2: 1

such that gn = 1. The group G is torsion-free if it has no torsion elements.

The following lemma is due to Karrass, Magnus and Solitar [KMS60]. A pro of

can be found in [LS].

Lemma 9.1 If G* is an HNN extension of the group Gand 9 E G* is of finite order,

then 3h E G of finite order su ch that 9 is a conjugate of h.

Proof. If 9 E G*, let 9 = htêlhl ··· ten be one of its cyclically reduced conjugates.

If m 2: 1, we have gm = hte1 hl ... ten htêl hl ... ten ... hte1 hl ... ten =1= 1 "lm E Z.

Thus if 9 is of finite order, then n = 0 and so 9 = h E G, but 9 = k-lgk by

definition, so 9 = kgk- l and is thus a conjugate of an element of G of finite order .•

Lemma 9.2 Let G have the one relator presentation (aI, ... 1 r), where r is cycli

cally reduced. If r is not a praper power then G is torsion-free.


Theorem 9.3 Let G have the one-relator presentation (t, aI, ... 1 rn ), where r is

cyclically reduced, not a proper power and n > 1, then 0 (r) = n and for any element

9 E G there is an integer m such that 9 is a conjugate of rm .

Proof. If r involves the single generator t, then r = t since it is not a proper

power, and thus G = (t 1 tn) * (al, ... 1 - ).

If 9 E G then 9 has one of the following forms:

gatn!gl'" tnkgk,

t no gl t n! ... gk t nk ,

gatn! gl ... gk_l tnk ,

t no gl t n! ... t nk-! gk·

Without loss of generality, say gis of the form gatn! gl ... gk_ltnk since the other cases

are similar. If k ~ 1 we have

sinee it is a normal form, thus if 9 is of finite order k = 0 must be the case. But

this means that 9 E (t 1 tn) or 9 E (al, ... 1 - ). The latter cannot be true since no

element of a free group is of finite order, thus 9 E (t 1 tn), which means 9 = tm for

sorne integer m.

80 we may assume that rn involves at least two generators, say t and a in G =

(t, a, bl , ... 1 rn) where r involves the generators t, a, bI, ... ,bk for sorne k, as our

usual convention. We proceed by induction on Irl.

Case 1: O't (r) = 0


By Theorem 3.2 (page 19), Gis the HNN extension of

where s = r' and r' is the rewritten form of r. In addition Ir'i < Irl.

Now if 9 E G is of finite order, then by Lemma 9.1 (page 46) 3h E H of finite

order such that 9 =G p-Ihp. But h is a conjugate of (r,)m for sorne integer m, so

there is q such that h =H q-l (r,)m q.

So 9 =G p-Iq-l (r,)m qp =G (qp)-l (r,)m (qp), and sinee r' -G r then 9 -G

(qp)-l r m (qp).

Case 2: No generator appears in r with exponent sum O.

As before '!jJ : G "-7 C = (y, X, bl , ... 1 s (y, X, bl , ... ,bt )) is an embedding and

C = '!jJ (G) * (x 1 - ). Here s = ri where rI is the rewritten form of rand (b)~(x"')

Irll < Irl·

So by induction, any element of C of finite order is a conjugate of rI for sorne

integer m. Now rI is a conjugate of r (yx- f3 , x(\ bl , ... , bn) = '!jJ (r), since it is obtained

from it by cyclic permutations. Thus any element of C, and henee of'!jJ (G), of finite

or der is a conjugate of '!jJ (rm).

Claim: If g, h E G such that '!jJ (g) = p-l'!jJ (h) p for sorne p E C, then there is

q E G such that 9 = q-Ihq.

Since C = '!jJ (G) * (x 1 - ), write p in the normal form as p = CPI .. , Pn and (b)~(xa)

proceed by induction on n.

If n = 0, then k E (b) ç '!jJ (G), so there is q E G such that '!jJ (q) = k and thus

'ljJ (g) = p-l'ljJ (h) p = 'ljJ (q-Ihq), and thus 9 = q-Ihq since 'ljJ is injective.

Now let p = CPI ... Pn with n arbitrary, then

'!jJ (g) p-l'!jJ (h) p = p;;l ... P1IC-I'!jJ (h) CPI ... Pn

- p;;l ... p11'!jJ (h) Pl" . Pn


and we have two cases: either kl E 'ljJ (G) or kl E (x 1 - ).

If Pl E 'ljJ (G), there is Pl E G such that kl = 'ljJ (Pl) and if we set ho = pllhpl' we

have

for sorne Co in the amalgamated subgroup. Since Po = P2 ... Pn has less syllables than

P then induction applies and the conclusion of the claim holds.

On the other hand, if Pl E (x 1 - ), let Pn, . .. ,Pl, Po be coset representatives of

P;;' 1 , ... 'PlI, 'ljJ (h) respectively.

for sorne Cl in the amalgamated subgroup. But by the normal form theorem, this can

only hold if 'ljJ (h) E (xCl<), or 'ljJ (h) = xnCl< for sorne integer n. But since Pl E (x 1 - ),

then there is an integer m such that Pl = xm, and thus Pll'ljJ (h)Pl = x-mxnO<xm =

xnCl< = 'ljJ (h), and hence

and induction applies as above. This ends the proof of the claim.

Now if 9 E G has finite order, then so do es 'ljJ (g), but we showed above that this

would mean that 'ljJ (g) is a conjugate of 'ljJ (rm ) for sorne integer m. Hence the claim

assures that 9 is a conjugate of rm as desired. _

9.2 n-freeness

There are many ways to generalize torsion-freeness. We notice that in a torsion

free group every subgroup generated by a single element is free. For instance, Pride

[Pr77-2] showed that if H is a two generator torsion-free subgroup of a one-relator


group G with torsion, then H is free.

A natural way to generalize torsion-freeness is to pass to n-freeness.

A group G is n-free if whenever gl, ... ,gn E G, we have (gb ... ,gn) ::; G is free.

This implies that any set of at most n elements generate a free subgroup.

Kapovich and Schupp [KSc04] proved that if the finitely generated one-relator

group with torsion, G = (ab' .. ,an 1 rffi) satisfy

m ~ M (6k - 2) + 2,

where M = max {aai (r) Il::; i::; n}, then Gis k-free.

In Section 11 we talk more about n-freeness from the point of view of cyclically

pinched one-relator groups.

9.3 Virtually Torsion-Freeness

A group G is virtually torsion-free if it contains a subgroup of finite index which is

torsion-free.

A group G is potent if for every nontrivial x E G and for every integer n > 0,

there is an epimorphism <p ; G ~ G' , onto a fini te group G' , such that 0 ( <p (x)) = n

in G' .

We remark that potency is a stronger property than residual finiteness.

Stebe [St71] showed that free groups potent.

Fischer, Karrass and Solitar showed that one-relator groups with torsion are vir

tually torsion-free, see [FKS72].

We prove this using potency.

Consider the group G with the one-relator group with torsion presentation (al, ... an 1 rn),

where r is not a proper power. Since free groups are potent, there is an epimorphism

<p ; F ~ G' where F = (al, ... an 1 - ) and 0 (<p (r)) = n. This induces a homo-

r-..


morphism cp : G -+ G'. The kernel ker <p of <p has index [G' : ker <pl < 00. Moreover

ker <p is torsion-free. lndeed any nontrivial torsion element 9 E G is a conjugate of

,m for sorne 0 < m < n by Theorem 9.3, but <p(,m) =/-1G' since o(<p(,)) = n. Thus

,m t/:. ker 4> so 9 t/:. ker <p.

9.4 Newman's Spelling Theorem

This theorem was first announced by Newman in [Ne68]. His proof, however, can

only be found in his Ph.D. thesis.

Theorem 9.4 Let G = (t, a, b1 , . .. 1 ,n), whe,e , is cyclically ,educed and n > l.

If W =0 v, whe,e w is reduced on G and v .omits a genemto, appearing in w, then w

con tains a subword s which is a subwo,d of r±n and

Remark 9.5 Note that when n = 1, Theorem 9.4 is a version of the Freiheitssatz.

The proof given below can be found in [L8].

Proof of Theorem 9.4. If w =0 v and v omits t, without loss of generality,

and t appears in w but not in " let K = (a, b1 , b2 , •• . )c; then G = K * (t)c and

v E K.

Write w = Wl ... Wk in the normal form, then there must be 1 ::; j ::; k such that

Wj = t m and m =/- O.

By the normal form theorem if w =0 v then there must be 1 ::; i ::; k such that

Wi =K 1.

Nevertheless, if the theorem holds for Wi =K 1 then it must hold for w =0 v.

80 we only consider cases when the omitted generator appears in ,.

We proceed by induction on 1,1. If, involves only one generator, then 3m such

that , = t m , then the omitted generator is t by assumption, since t appears in w and


W =a V which omits t. Then we can only achieve this if w has the subword tpmn

where p =1= 0, since tmpn = 1 Vp =1= O. In which case Itpmnl = Ipl n Iml > (n - 1) Iml =

(n - 1)/ n Irnl. 80 we may assume that r involves at least two generators. As is our usual con-

vention we say that r involves t, a, b1 , ... , bk , 0 :::; k < 00.

Case 1: sorne generator occurring in r has zero exponent sumo

We may relabel generators so that t is this particular generator. Thus at (r) = 0

and so G is the HNN extension of H whose presentation is

In addition,

are both freely generated in H.

If, on the one hand, w E W (t, a, b1 , ... ) is reduced and w (t, a, bI, ... ) = v (a, b1 , ... )

where t occurs in w but not in V.

Consider the foUowing operation:

Replace every subword têu (a1m1, ••. , al

M1, ••• , akmk , ... , akMk' ak+l, ... ) t-ê, where

[ = ±1, by u(alml+e, ... ,alMl+e, ... ,akmk+e, ... ,akMk+e,ak+l+ê,"')' where aU the

generators involved in u are the generators of X whenever [ = 1 or Y whenever

[ =-1.

Now st art with w and perform the above operation repeatedly to get the word

w', where we can no longer perform the operation.

There are two cases either w' involves t or not. If w' involves al then w' contains

a subword têuCê, and u does not involve t and is not a word on X if [ = 1 or on Y


if é = -1, but U =H z.

But then Z omits a generator of H involved in u and u =H z, thus by induction

u contains a subword s which is a subword of p±n such that Isl > (n - 1)/ n Ipnl.

Now recover w from w' be replacing each Xi by tixori and reducing by cancelling

only t's.

Denote the part recovered from s by s', then s' is a subword of r±n and 1 si>

(n - 1)/ n Irnl as desired.

If we obtain a word w* from 11) which does not involve t, then w* must involve

a generator of H of nonzero subscript and w* =H V. By induction w* contains a

subword s of r±n and may be chosen such that it does not begin or end with t±l.

If, on the other hand, w (t, a, b1 , ... ) =0 v (t, b1 , ... ), where a occurs in w but not

in v, then at (w) = at (v) =: a, and so wt-Œ =0 vrŒ. Since at (vrŒ) = 0 we can

rewrite vrŒ by our operation into v* which does not involve t, and so wt-a =0 v*.

Now as in the previous case if wr a cannot be reduce to a word not involving t and

thus it contains a subword s which is a subword of r±n such that Isl > (n - 1)/ n Irnl

and s does not begin or end with t±1. This ensures that s is a subword of w not of

wrŒ•

If wt-a can be reduced to a word w* not involving t then sorne aij occurs in w*. But

since w* = H v* then w* contains a subword s'of p±n such that 1 s' 1 > (n - 1) / n Ipn 1

and thus w contains a subword s of r±n such that Isl > (n - 1)/ n Irnl and s does

not begin or end with t±l.

Case 2: AH the generators appearing in r occur with nonzero exponent sumo

If w (t, a, b1 , ... ) = v (a, b1 , ... ), let a .- at (r) and j3 := aa (r) then 'ljJ : t f--7

yx-{3, a f---+ x a , bi f--7 bi is an embedding


Freely redueed w (yx- f3 , x Œ, Cl, ... , Cn) to w' and v (x Œ

, Cl, ... , Cn) to v'. Here

y appears in w' but not in v'. Since (J x (r (yx- f3 , x Œ, Cl, ... , Cn)) then by Case 1

w' contains a subword s' which is a subword of r±n (yx- f3 , x Œ, Cl, ... , Cn) such that

Is'I> (n-1)/n Irn (yx- f3 ,xŒ ,cl, ... ,cn)1 and s' does not begin or end with x±l,

rewriting s'as s using yx- f3 I--t t, x Œ I--t a, bi I--t bi we get Isl > (n - 1)/ n Irnl. •

Corollary 9.6 The subword s of Newman's spelling theorem is of the form Tn-ITo,

where T is a cyclic rearrangement of r±l, and To is a proper initial segment of T.

Proof. Since s is a subword of r±n, then r±n = tsu, where t, u are subwords,

possiblyempty, and Isl 2: Ir±nl·

Now

so

Consequently, t and u are subwords of r (when s is a subword of r n ) or of r-1(when

s is a subword of r-n ).

Henee r±l = tto or r±l = uou (treating both cases simultaneously), so

and thus

±n ±l ±l tt tt r = r ... r = o··· ouou ~~

n-times (n-l)-times

s = tot ... tottouo ~ (n-l)-times

since tsu = r±n = t (to ... ttouo) u from (9.1).

(9.1)

Setting T = tot and To = touo, we get from (9.1) that s = Tn-1To, and T is a

cyclic rearrangement of r±l .

9 ONE-RELATOR CROUPS WITH TORSION 55

In addition

Tto = totto = tor±l = touou = Tou.

However, ITol < ITI, so To is a subword of T and is, in fact, an initial segment of

T .•

10 COMMUTATIVITY OF ONE RELATOR GROUPS 56

10 Commutativity of One Relator Groups

In this section we describe sorne properties of Magnus subgroups of one-relator groups.

We start with two simple results on free products.

Proposition 10.1 If C = A * Band 9 ~ A, then g-1 Ag n A = (1).

Instead of proving this proposition we prove the slightly more general Proposi-

tion 10.2.

Proposition 10.2 If C = A *c Band 9 ~ A, then g-1 Ag n A ~ C.

Proof. Since 9 E C, then write it in the unique normal form, 9 = cgl ... gn'

If x E g-1 AgnA, then x E A and:la E A such that x = g-lag. But x, a E A, thus

:lCl, C2 E C and coset representatives a', x' of AI C such that x = CIX' and a = C2a',

and this is the normal form of x and C respectively.

Now

-1 -1 -1 -1 -1 -1 - gn ... gl C acgl'" gn = gn ... gl C cagl'" gn

-1 -1 - gn .,. gl agI'" gn

If, on the one hand, gl E A, find coset representative Ci of g11ag1 E A and {Ji of

g;\ i = 2, ... ,no Then

-1 - - -x = 9 ag = gn ... g2ag2 ... gn

is a normal form for X.

If, on the other hand, gl E B, find coset representatives ?Ji of g;1, i = 1, ... ,n.

Then

-1 - - -x = 9 ag = gn ... g1 agI' .. gn

is a normal form for X.


In either case, we get a contradiction to the uniqueness of the normal form unless

a E C, in which case by commutativity of elements of A and B with elements of C

we get

and thus g-l Ag nA::; C. •

In his paper, Newman [Ne68] announced the following theorem that he had proved

in his Ph.D. thesis.

Theorem 10.3 (Newman) If G has the one relator presentation (X 1 rn ) with tor

sion and L ç X, then 9 tJ. L := (L)G' then g-l Lg n L = (1).

We are not going to prove this theorem, but instead we are going to prove a

generalization provided by Collins. Both Bagherzadeh [Ba76] and Collins [Co04]

generalized this theorem to Magnus subgroups of any one-relator groups.

Theorem 10.4 (Bagherzadeh) Let G have the one-relator presentation (X 1 rn)

and M ::; G a Magnus subgroup, then Vg tJ. M, g-l Mg n M is cyclic.

As an illustration we shall prove it when the relator r involves at most two gen

erators. We follow the proof in [Ba 76].

Proof. Case 1: r is empty, i.e. Gis free.

If M = (L)G is a Magnus subgroup, L ç X and G = (X 1 - ), then either L = X

or LeX. If L = X, there is no 9 E G" M and thus the conclusion holds. If, on

the other hand, LeX, then if M' = (X " L), we see that G = M * M', the free

product oftwo Magnus subgroups. Now if 9 tJ. M, then by Proposition 10.1 (page 56)

g-l Mg n M is trivial and is thus cyclic.

Case 2: r involves a single generator, or, in other words, r = am for sorne integer

m and generator a.

la COMMUTATIVITY OF ONE RELATOR GROUPS 58

We see that

Now M = (L) and L must omit a, the only generator appearing in r, then M :::; Ma.

Let 9 E G" M be given, then either 9 E Ma" Mor 9 E G" Ma. If 9 E Ma" M the

conclusion holds by the first observation, and if 9 E G" Ma, then Proposition 10.1

(page 56) guarantees that

and so g-l Mg n M is cyclic.

Case 3: r involves two generators a and b.

In this case

G = (X 1 r) = (a, b 1 r) * (X" {a} 1 - ) = (a, b 1 r) * Ma (b) (b)

and M :::; Ma as in Case 2. Let 9 E G" M be given, then either 9 E G" Ma, and

thus Case 1 takes care of the pro of, or 9 E Ma " M, and thus by Proposition 10.2

(page 56)

and is thus cyclic as desired. _

What remains is to prove the theorem for generators involving at least three

generators. The proof proceeds by induction and follows the pattern of the proof of

the Freiheitssatz, in which we embed G in an HNN extension of another one-relator

group with shorter relator. The proof involves properties of commutators of Gand

can be found in Bagherzadeh's paper [Ba76].

The following is Collins' generalization of Newman's theorem.


Theorem 10.5 (Collins) Let G = (X 1 rm), r being a cyclically reduced relator

with m ~ 2. If M, N :::; Gare any two Magnus subgroups and 9 rj:. N M, then

g-IMgnN=(I).

Proof. If there is 9 rj:. MN such that g-IMg n N =1= (1), then ::lu E M,v E N

such that g-lug = v. Pick u, v among all such possible choices such that 9 is of a

minimallength. This means that g-lug is reduced, for if it were not reduced, then 9

would not be reduced, and if we reduce it we get a shorter go such that gr;lugo = v,

contradicting minimality.

Now g-lumg = (g-lug)m = vm for any integer m, thus we may assume the words

in M and N representing u and v, respectively, are of length greater than r.

So 9 rj:. MN, u E M, vEN and g-lugv =0 1, thus by Newman's spelling theorem

there is a subword tm-Itl such that t is a cyclic rearrangement of r±l and tl is an

initial segment of t.

Choosing u, v very long we may ensure that tm-Itl is a subword of either g-lu, ug

or gv-l . Without loss of generality, say that tm-Itl is a subword of either g-IU.

Since u omits a generator occurring in r, then a nontrivial part of tl, say t~, lies

in g, so tm-It~ is a subword of g. But since t~ is an initial segment of t then ::lt~ such

that t = t~ t~, thus tm-

l t~ t~ = tm, which is a cyclic rearrangement of rm =0 1, and

Irl = Itl = It~1 + It~l, which implies that It~1 < Irl since t~ is nontrivial. So 9 = stm-It~

and we may replace tm-It~ by t;-l, since tm-lt~ t~ =0 1, to get a presentation of 9 by

st;-l of shorter length-contradiction. _

To conclu de we state a theorem proven recently by Collins in [Co04J.

Theorem 10.6 Let G be a one-relator group with the two Magnus subgroups M =

(X)G and N = (Y)G' then Mn N = (X n Y)G or (X n Y)G * Z.

11 THE ISOMORPHISM PROBLEM 60

Il The Isomorphism Problem

In Section 7 we discussed Pietrowski's solution of the isomorphism problem for one

relator groups with nontrivial centers. In addition Pride [Pr77-1] showed that the

isomorphism theorem is solvable for two-generator one-relator groups. Sela [Se95]

proved that the isomorphism problem is solvable in torsion-free word hyperbolic-

groups.

A geodesic metric space has 6-thin triangles if for any geodesic triangle ab, bc, ca

we have ab E N8 (bc U ca).

A finitely generated group G is word-hyperbolic if the Cayley graph of G has 6-

thin triangles. Notice that this do es not depend on the choice of the finite generating

set. We refer the interested reader to [ABCFLMS] or [GR] for an introduction to this

subject.

Note that finitely generated C' (1/6) groups are word-hyperbolic, for instance

consult [GR]. Thus most one-relator groups are word-hyperbolic as in Section 4.

Even though Sela's result is strong, unfortunately there is no known algorithm to

de ci de if a one-relator group is not word-hyperbolic.

Let m, n be such that m 2:: 2 and n 2:: 1. Consider the class of groups with

presentations (al, ... , am 1 rl, ... , r n) such that the r/s are cyclically reduced and

Iril ::; t. Let N (m, n, t) be the number of such presentations. If P is a property, let

Np (m, n, t) be the number of presentations from the above class satisfying P, then

P is exponentially (m, n )-generic if

exponentially fast.

Np (m, n, t) ~ 0 N (m, n, t) t->oo

Now if m 2:: 2 Kapovich and Schupp [KSc05] showed that there is a class Cm of

m-generator one-relator groups such that belonging to Cm is an exponentially (m, 1)-

11 THE ISOMORPHISM PROBLEM 61

generic property and the isomorphism problem is solvable in exponential time for

pairs of groups in Cm.

12 TORSION-FREE ONE-RELATOR GROUPS 62

12 Torsion-Free One-Relator Groups

From Theorem 9.2 (page 46) we know that a one-relator group G = (al,'" 1 r)

is torsion-free if and only if r is not a proper power. In this section we are going

to explore some special torsion-free one-relator groups, namely, surface groups and

cyc1ically and conjugacy pinched one-relator groups.

Definition 12.1 The fundamental group of a compact surface of genus n is a surface

group.

A surface group of a compact orientable surface of genus n 2: 2 is

(12.1)

and that of a compact nonorientable surface of genus n is

(12.2)

These groups were the first examples of one-relator groups and have been heavily

studied.

It is known for surface groups that if gl, kl, ... , gn-l, kn-l, gn E Gn, then (gl' k l , ... , gn-l, kn-l

Gn is free. Similarly, if hl, . .. ,hn-l E Hn, then (hl, ... ,hn- l ) ~ Hn is free.

We are naturally looking for generalizations of surface groups that would have

similar resul ts.

Definition 12.2 G = (Xl, ... ,Xn 1 u (Xl, ... ,xp) = V (Xp+l' ... ,xn)), 1 < p < n, is a

cyclically pinched group.


If we set U (Xl, YI) = [Xl, Ylrl

and v (X2, Y2, ... , Xn, Yn) = [X2, Y2]' .. [xn, Yn] in the

above definition, we see that the presentation (12.1) is

Gn = (Xl, YI, ... ,Xn,Yn 1 U = v)

and is thus cyclically pinched. Similarly if we set U (Xl) = x12 and v (X2, ... , x n ) =

x~ ... x~" we see that the presentation (12,2) is

Hn = (Xl, ... , Xn 1 U = v)

and is thus cyclically pinched.

Rosenberger [Ro81] proved the following theorem.

Theorem 12.3 If G = (Xl,"" Xn 1 U (XI, ... , Xp) = V (Xp+l,"" xn)), and neither u

nor v is a proper power in (Xl, ... ,xp 1 - ) and (xp+1, ... , Xn 1 - ) respectively, then

G is 3-free.

Baumslaghad proved that in the same settings Gis 2-free.

We may extend the theorem to the following theorems that are found in [FGRS93].

A word w E G = (X 1 R) is primitive if w E X.

Theorem 12.4 Let Xl,"" X n be pairwise disjoint sets of generators such that IXil 2::

2, i = 1, ... , n, and Wi = Wi (Xi)' i = 1, ... , n, be nontrivial words on (Xi 1 - ) that

are not proper power and not primitive. If

then G is n-free.

If we relax the hypothesis regarding proper powers, we get the following result.


Theorem 12.5. Let Xl, ... ,Xn be nonempty pairwise disjoint sets of generators, and

Wi = Wi (Xi) E (Xi 1 - ), i = 1, ... , n, be nontrivial words that are not primitive. If

then G is (n - l)-free.

The importance of cyclically pinched one-relator groups is not limited to this.

In fact, Baumslag [Bau85] showed that they are residually finite, Lipschutz [Li75],

by means of small cancellation theory, showed that they have a solvable conjugacy

problem, Rosenberger [Ro94] showed that they have a solvable isomorphism theorem.

The following theorem is deduced by Fine, Rosenberger and Stille [FRS97].

Theorem 12.6 If G = (Xl,"" Xn l 'U (Xl, ... , Xp) = V (Xp+l,"" x n )), and neither 'U

nor v is a proper power in (XI, ... ,xp 1 - ) and (Xp+l, ... ,Xn 1 - ) respectively, then

G is word hyperbolic.

A Baumslag-Shalen decomposition of a one-relator group G is a decomposition of

the form G = A *c B, where A, B, and C are finitely generated.

Fine and Peluso [FP99] proved the following remarkable theorem.

Theorem 12.7 Let G be a torsion-free one-relator group with Baumslag-Shalen de

composition G = A *c B, with both A and B free. If either of [A: C] < 00,

[B : C] < 00, C = [A, AL or C = [E, E] is true, then G is cyclically pinched.

Definition 12.8 If 'U, V are words on {Xl,"" Xn}, then

is a conjugacy pinched group.

12 TORSION-FREE ONE-RELATOR GROUPS

If we set t = Yn, U = Xn and v = [XI, YI]' .. [xn-I, Yn-l] Xn in (12.1), we get

G / 1 -1 -1 -1 -1 1\ n - \Xl, YI, : .. ,Xn, Yn XlYlX l YI ... XnYn Xn Yn = /

and consequently surface groups are conjugacy pinched groups.

65

We would like to show that conjugacy pinched one-relator groups have properties

similar to those of cyclically pinched one-relator groups.

A conjugacy pinched one-relator group G = (Xl, ... , Xn 1 turl = v) is generic if

u, v are not proper powers in (Xl, ... ,Xn 1 - ).

The following theorem is due to Fine, Rohl and Rosenberger [FRR93].

Theorem 12.9 If G = (Xl, ... , Xn 1 turl = v) is a generic conjugacy pinched one

relator group, then (x, y) :::; G is either free of rank 2, abelian, or has presentation

(a, b 1 aba-l = b-l ).

The same authors tried to generalize their result in [FRS97-2], they consequently

proved the following two theorems.

Theorem 12.10 IfG is a generic conjugacy pinched one-relator group such that u is

not a conjugate to either v or V-l, then (x, y, z) :::; G is either free of has a one-relator

presentation.

A subgroup M = (x, y) :::; Gis maximal if whenever M :::; (z, w) then M = (z, w).

A maximal subgroup M = (x, y) :::; G is strongly maximal if Va E G ::lb E G such

that (x, aya-l ) :::; (x, byb-l ) and (x, byb-l ) is maximal.

The following theorem is stated in [FRS97 -1].

Theorem 12.11 If G = (Xl, ... ,Xn 1 turl = v) is a conjugacy pinched one-relator

group such that (u, v) :::; (Xl' ... ,Xn 1 - ) is strongly maximal, then G is 3-free.


The isomorphism theorem has a partial solution in conjugacy pinched one-relator

groups such as in the following theorem found in [FRS97-1].

Theorem 12.12 If G is a generic conjugacy pinched one-relator group, and there is

no Nielsen transformation from {Xl, ... , Xn} to a system {YI, ... , Yn} such that U E

{YI, ... , Yn}, and no Nielsen transformation to {Zl' ... , zn} such that V E {Zl' ... , Zn},

then the isomorphism problem is solvable for G.

A group Gis subgroup separable if for any finitely generated H :::; Gand 9 E G ........ H

there is N <J G such that gN n H = 0.

Wise proved the following.

Theorem 12.13 Let G = (al,"" an, tl,"" tm 1 t l u l t 11 = VI,···, tmUmt":;;/ = Vm),

where UI, VI, ... , Um, Vm are non trivial cyclically reduced words on (al, .. . ,an 1 - ).

If G does not contain a subgroup isomorphic to

the Baumslag Solitar group, with n =J ±m, then G is subgroup separable.

Two elements x, y EGare conjugacy separated iffor any t, s E G, (ut)n(v S) = (1).

Finally Fine, Rosenberger and Stille [FRS97-1] stated the following theorem.

Theorem 12.14 If G is a generic conjugacy pinched one-relator group and u, v are

conjugacy separated in (Xl, ... ,Xn 1 - ), then G is word hyperbolic.

13 A5PHERICITY 67

13 Asphericity

For homotopy background we direct the reader to [Ha]. Most of the definitions below

can be found in [BP93].

A 2-complex is aspherical if any map 52 ~ X is null-homotopic, i. e. homotopie

to a point. A presentation G = (X 1 R) is aspherical if its standard 2-complex is.

In the presentation G = (X 1 R), if Rand X o ç X are linearly ordered and r E R,

we set

minr - min {x 1 x appears in r and xE X o}

maxr - max{x 1 x appears in r and x E X o}

The presentation G = (X 1 R) is staggered if Rand X o ç X are linearly ordered

such that:

1. 'IIr E R, r is cyclically reduced and involves sorne x E X o.

2. If r, ro E R such that r < ro, then min r < min ro and max r < max ro

Clearly, any one-relator group is staggered with X o = X = {XI, .. . ,xn } ordered

linearly as Xl < ... < X n .

Karrass and Solitar [KS71] proved the following theorem.

Theorem 13.1 If the presentation G = (X 1 R) is staggered, then it is aspherical.

So, in particular, one-relator groups are aspherical.

A map 'l/J : X ~ Y between CW complexes is combinatorial if for any cell C in

X 'l/J 1 c : C ~ C' is a homeomorphism from C to a cell C' in Y.

A spherical diagram is a combinatorial map 8 2 ~ X into a CW complex X.

A pair of 2-cells C, C' in a spherical diagram D ~ X, meeting along an edge

e, is cancellable along e if 8C and 8C' starting at e in the same direction are such

13 ASPHERICITY 68

that 'ljJ (aG) = 'ljJ (aG') in X. X is combinatorially reducible if every diagram has a

cancelable pair.

Lyndon [Ly50] proved the following theorem.

Theorem 13.2 Every staggered 2-complex is combinatorially reducible.

A proof of Theorem 13.2 can be found in [L8]. Hruska and Wise [HW01] provided

a proof using towers and ladders.

In particular, every one-relator groups is combinatorially reducible.

Bogley and Pride [BP93] define a presentation G = (X 1 RI to be diagrammati

cally reducible if it is combinatorially reducible, none of the relators in R is a proper

power, and no ri is a cyclic permutation of rj1's. Thus a torsion-free one-relator

group is diagrammatically reducible.

14 EXPONENTIAL GROWTH 69

14 Exponential Growth

To define what it means for a group to have exponential growth, we need to define a

chain of mappings as follows.

Given any group G generated by a finite set S, we define the growth function

(Js : N -+ N such that

(Js(k) = #{w E Glls(w) ~ k}

where ls is the length function associated to the generating set S.

The exponential growth rate is defined as

w(G, S) = limsup\!(Js(k) n~oo

and then we set w(G) := inf {w(G, S)I G = (S), ISI < oo}.

Now G has exponential growth if w(G, S) > l for some finite generating set S,

and has uniform exponential growth rate if w( G) > 1.

The natural question that might arise is whether there is a group having expo

nential growth but not uniform exponential growth. The question is still open, but

the answer is in the negative for any one-relator group. The following theorem due

to Grigorchuk and Harpe [GrHOI] should take care of the question.

Theorem 14.1 Any one-relator group of exponential growth has uniform exponential

growth. Moreover, we have w( G) ~ V"2

/"--

15 ONE-RELATOR PRODUCT 70

15 One-Relator Product

In this section we survey results on one-relator products.

Definitions of this section can be found in [Ho84].

Given a cyclically reduced word r E A * B, then the group G = A * BI ((r)) is

called the one-relator pro du ct of A and B.

It is worthwhile to notice that if A and B are both free, say with presentation

A = (ai;i E 1 1 - ) and B = (bj;j E J 1 - ) for sorne index sets 1 and J, then

A * BI ((r)) has the presentation (ai, bj ; ; i E 1, j E JI r), which is a one-relator group,

and we obviously see that one-relator product generalizes one-relator groups.

In general if {Ai}iEI is a farnily of groups with r E *iEIAi is cyclically reduced,

then their one-relator product is

We will adopt the notation AJ = *iEJAil ((r)) for any J c 1, and thus G can be

denoted by Ad ((r)).

A group G is locally indicable if for any finitely generated subgroup H ~ G there

is an epirnorphisrn ~: H ~ Z.

The following theorern is proven by Brodski! [Br81] and reproven by Howie [Ho84].

Theorem 15.1 Torsion-free one-relator groups are locally indicable.

Unfortunately, the Freiheitssatz. does not generalize to one-relator products of

any groups; however, Brodski! generalized it to locally indicable groups [Br84].

Theorem 15.2 (Brodski'l) Let J ç 1 be given. 1fr is not a conjugate of an element

of AJ, then AJ ~ Ad ((r)).

We will calI ~ (AJ) ç Ad ((r)) a Magnus subgroup.


Freiheitssatz. is not the only theorem that is generalized to one-relator products,

we also have Newman's theorem of the intersection of two Magnus subgroups, see

[Ho89].

Theorem 15.3 (Howie) Let {A}iEl be a family of locally indicable groups. If G =

Ad ((r)), n> 1 and M, N ç 1, then AM n AN = (1).

The following is a generalization of Collin's theorem, see [Ho89].

Theorem 15.4 (Howie) Let {Ai}iEl be a family of locally indicable groups. G =

Ad ((r)) and M, N ç 1, then AM n AN = AMnN * (1), or AMnN * Z.

In addition, Howie gives more conditions to determine when the free factor is

trivial or infinite cyclic, and provides an algorithm to determine it.

If we do not impose conditions on the factor groups, and restrict the relator

instead, we get strong results, see [Ho89].

Theorem 15.5 (Freiheitssatz) Ifr is cyclically reduced such that Irl 2: 2 and m 2:

4, then for any J ç 1, AJ '-' Ad ((rm))

Theorem 15.6 If r is cyclically reduced such that Irl 2: 2 and m 2: 4, then for any

J ç l and the world problem is solvable in Ai 'Vi E 1, then so it is in A J '-' Al/ ((rm)).

A relator rm is of the form E (p, q, m) if there are letters x and y whose orders

are p and q respectively, and a word u such that r = xu yu- l .

The subwords UI, ... ,Uk are strongly disjoint if there is a permutation 0" and

nonempty subwords VI, ... ,Vk such that w = Ucr(I)VI .•. Ucr(k)Vk.

Now we state two spelling theorems for one-relator product A * B / ((rm )) due to

Duncan and Howie [DH94].

Theorem 15.7 Ifrm is not of the form E (2, 3, 4) nor E (2, 3, 5), let w E ((rm)) be a

nonempty cyclically reduced word, then one of the following is true:


1. w is cyclic permutation of r±m.

2. w contains two strongly disjoint subwords U1 and U2 which are subwords of r±m

and such that IUil ;::: Irm-

11- 1, i = 1,2.

3. w contains strongly disjoint subwords U1, ... ,Uk, 3 ::; k ::; 6 which are subwords

of r±m and su ch that IUil ;::: Irm-

21 - 1 if i ::; 6 - k and IUil ;::: Irm

-3

1 - 1 if

i > 6 - k.

Theorem 15.8 If r has no letier of order 2 in A or B, m ;::: 3 and w E ((rm)) is a

nonempty cyclically reduced word, then one of the following is true:

1. w is cyclic permutation of r±m.

2. w contains two strongly disjoint subwords U1 and U2 which are subwords of r±m

and such that IUil ;::: Irm-

1 1, i = 1,2.

3. w contains three strongly disjoint s'Ubwords U1, U2, U3 which are subwords of r±m

and su ch that IUil ;::: Irm-

2L i = 1,2,3.

Brodoskir generalized Murasugi's results on the Centre of one-relator groups to

one-relator produets of loeally indieable groups. Namely, if A and B are loeally

indieable and C ::; A * B / (( r m)) sueh that Z (C) =1 (1), then Z (C) is infinite eyclie.

In addition if Z (C) is nontrivial, and A (respeetively, B) is noneyclie, then Z (C) ::; A

(respeetively, ::; B); see [MeOl].

16 OPEN PROBLEMS 73

16 Open Problems

The following is a list of selected open problems. We were highly influenced by [FROl].

1. The isomorphism problem for one-relator groups. In fact this is even open for

one-relator groups with torsion.

2. B.B.Newman showed that the conjugacy problem for one-relator groups with

torsion is solvable, see [Ne68]. More generally it is solvable for torsion-free

word-hyperbolic (one-relator) groups, see [GR]. No solution is known for the

conjugacy problem in general.

3. ls the generalized word problem solvable for one-relator groups? i. e, is there an

algorithm for deciding if a given element of the group belongs to a given finitely

generated subgroup?

4. (Gersten) ls every one-relator group without Baumslag-Solitar subgroups hy

perbolic?

5. ls every one-relator group without BS (n, m) ,n =f ±m, automatie?

6. (G.Baumslag) Are aIl one-relator groups with torsion residually finite?

7. (C.Y.Tang) Are all one-relator groups with torsion conjugacy separable?

8. (G.Baumslag) Let H = FI R be a one-relator group, where R is the normal

clos ure of an element r in F. Then, let G = FIS be another one-relator group,

where S is the normal closure of s = rk for sorne integer k. ls G residually finite

whenever His?

9. (G.Baumslag) Let G rv (al, ... 1 lu, v]). ls G residually finite?

10. (D.Maldavanskil) Are two one-relator groups isomorphie if each of them is a

homomorphie image of the other?

16 OPEN PROBLEMS 74

11. Are aH freely indecomposable one-relator groups with torsion co-hopfian?

(a) Which finitely generated one-relator groups have aH generating systems

(of minimal cardinality) Nielsen equivalent to each other ?

(b) Which finitely generated one-relator groups have only tame automorphisms

(i.e., automorphisms induced by automorphisms of the ambient free group)

?

12. (B.Fine) If Gis a one-relator group with the property that VH ~ G such that

[G : Hl < 00 then His a one-relator group, and V H ~ G such that [G : Hl = 00

then H is free. Must G be a surface group?

13. (J.Hempel) 1s every finitely generated normal subgroup of a finitely generated

one-relator group (with at least three generators) either offinite index or trivial?

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28 [Ro81] Rosenberger, G., On one-relator groups that are free products of two free

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32 [Ro94J Rosenberger, Gerhard, The isomorphism problem for cyclically pinched one

relator groups. J. Pure Appl. Algebra 95 (1994), no. 1, pp. 75-86.

25 [Se95J Sela, Z., The isomorphism problem for hyperbolic groups. 1. Ann. of Math. (2)

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REFERENCES 80

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ment. Math. Helv. 76 (2001), no. 2, pp. 314-338.

Index Commutator, 34

Concatenation, 8

Exponential

-Growth, 69

-Growth Rate, 69

Uniform-Growth Rate, 69

FormaI Inverses, 8

Freiheitssatz, 22

Group

Free--, 7

Nilpotent-, 34

One-relator-, 16

Positive-, 34

Presentation of a-, 11

Relators of a-, 11

Solvable-, 34

HNN

-Extension, 15

Product

Free-,15

Free-with Amalgamation, 14

One Relator-, 70

Tree-,42

Residually

81

-Finite, 36

Transformation

Deriving-,8

Nielson-, 13

Tietze-,12

Word, 8

Cyclically Reduced-, 10

Empty-,8

Exponent sum of a-, 10

Length of a-, 9

Reduced-, 10

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