THE INFINITY

Annales Academie Scientiarum Fennica Series A. I. Mathematica

Volumen t4, 1989, 129-131

ON THE INFINITY OF THE

LONERGAN_HO SACK PRES ENTATION

Markku Niemenmaa and Gerhard

Rosenberger

Introduction

In

1,984 F.D. Lonergan and J. Hosack _[3] introduced the following group ^pre- sentation:

G: ^{(r,z: z3*z3r-r} : zs12zzr'-

r).

According

to

the authors there was good reason

to

believe

that

the presentation is

that

of a finite group. However, they reported that attempts to prove finiteness by using the Todd-Coxeter algorithm were unsuccessful.

By using the computer algebra system CAYLEY

M.

Slattery [5] managed to show in 1985

that

G is infinite. To be precise, Slattery ^showed

that

G ^{has a} factor group which is infinite.

In this short note we wish to point out

that

a

lot

more can be said about the structure

of G.

In fact, we shall consider the presentation

G(n)

: _larz

:

znazn0-r :

zn*za222x2

:1)

with n ) 2. In

particular, our results hold

for G: ^G(3).

^{For the background}

material of this note the reader is advised

to

consult _[7].

Main theorem

Consider

the

presentation

G(n)

given

in

the introduction

with n ) 1. ^If n:7,

^then

zrz : n2z

and z5

: z-5: 14.

Consequently G(1) is a finite group of order

40.

Now we shall establish the

Theorem. Let n ) 2

and consider the presentation

G(n).

-lfow

(t)

G(rz) is

a

nontrivial free product

with

amilgamation,

(2) G(n)

^has

a

subgroup of frnite index mapping onto a free group of rank 2

and

G(n)

^has a free subgroup of rank 2,

(S)

G(n)

^{has a} generating pair

{u,u}

^{such that}

^the

^subgroup

(rk,rr)

is åee

of

rank 2 for a sufficiently large integer k,

(4) G(n)

is SQ-universal (i.e. every countable group is embeddable

in

some factor group

of

G(n)).

doi:10.5186/aasfm.1989.1422

130

M.

.lfiemen maa and G. Rosenberger

Proof. (1)

If n

is even, then the free product

Za* 22: (u,z

: n4

: ,2 :1)

is an epimorphic image

of G(n).If. n

is odd

(then n ) 3),

we have

D - (t,z:

zn

-

^(22x2)'

: 1)

as an epimorphic image

of G(n).

Now

D

^{is a} nontrivial free product of

är

and

Hz withthe

amalgamated subgrouP

ä,

^where

H1: (rl = Z,

Hz: (y,r, ," : ^(r2y)2 - ^{1) and} ä : l*') = fu). ^By

Lemma 3.2

of

_[6]

we conclude

thal G(n)

is a nontrivial free product

with

amalgamation for every

n)

^2.

(2) The free product

Za*

22 ha,s the triangle-group

fQ,4,5)-

(o,b i a2

-

^b4

-

⁽

ob)'-

¹⁾

as an epimorphic image and clearly G(n.) has

T(2,4,5)

as an epimorphic im- age provided

n

is even. Next consider

D

(the free product

with

amalgamation) introduced in the

first

part of the proof. Now

D

has the triangle-group

T(n,7,2) :

(o,b : an

-

^b7

-

^{( ob)'}

-

¹⁾

as an epimorphic image Thus

G(")

has as

and naturally

G(")

has

T(",7,2)

^{as an} epimorphic image an epimorphic image a triangle-group

T(p,

q,r) :

(a,b ; aP

:

bq

:

(ab)'

:

Ll

with 2 Sp,q,r

^and

(tld+Olil+Glr) ^<

¹

^.

^Now

^T(p,q,r)

^{has a surface group}

F(g)

of genus g

2 ²

^as a subgroup of finite index (see

[7]).

^Since

F(g) k 2Z)

has a free group of

rank 2

as an epimorphic image,

it

follows

by

considering the pre-image

ot F(g) in G(n) that G(n)

has a subgroup of finite index which maps onto a free group

ofrank

2 and consequentl1

G(")

has a free subgroup

ofrank

2.

(3) Since

T(p,q,r)

can be regarded as a subgroup of P,S.L(2,.E) ^(see [7]), our assertion follows from _[4].

(a)

BV [1] we know

that

a free group of

rank 2 is

SQ-universal. Clearly, ^a pre-image of an sQ-universal group is sQ-universal. Finally, by _[2]

it

follows that a group which has an SQ-universal subgroup of finite index is also SQ-universal.

The proof is complete.

Remarks.

^As indicated in the beginning of this paper G(1) ^{is a} finite group of order 40 and consequently

it

^has elements of finite order

) 2.

If we consider G(2) _, then

z2rzzr-r : z4r2z2a2:

1 implies o2z2

:

zza2

,

Ftrthermorer z6

: r-4and

2-6: r-4,

hence

zr2:L

^and

18: ^L.

Now we state

Problem 1.

^Is

it

true

that G(n)

has elements of finite order

) 2 fot n)

^3?

We also state

Problem 2.

Is

it

true

that G(n)

^{has a} torsion-free normal subgroup of finite

indexfor n)2?

On the

infinity

of the Lonergan-Hosack ^presel2 tation 131 References

tU

^HIGMAN, G., B.H. NsuulNN, and H. NpulrlNN: Embedding theorems for groups. - J.

London Math. Soc. 24, t949, 247-254.

l2l

^{NouMl.NN, P.M.: The} SQ-universality of some finitely generated groups.

-

J. Austral.

Math. Soc. L6, 1973, 1-6.

t3]

^{LoNnRGlN, F.D., and J.} Hos,q.cr: Problem 308.- Notices Amer. Math. Soc. 31, 5, 1984.

t4]

RospNspRopR, G.: On discete free subgroups of linea.r groups.- J. London Math. Soc.

(2) 17, 1978, 79-85.

t5]

Sr,nrreRy, M.: Solution to AMS Notices query No. 308. - Cayley Bulletin 2, 1985,37-39.

t6]

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t7]

ZrESCn.LNc, H., E.

Vocr,

and H.D. Colopwpv: Surfaces a.nd planar discontinuous groups.

-

^Lecture Notes in Mathematics 835. Springer-Verlag, Berlin-Heidelberg- New York, 1980.

University of Oulu University of Dortmund

Department of Mathematics Department of Mathematics

SF-90570 Oulu ^{D-4600 Dortmund 50}

Finland ^Federal Republic of Germany

Received 8 February 1988