18 - Generalised triangle groups of type (2, m, 2)

Summary

INTRODUCTION

A generalised triangle group is a group with presentation

where l, m, n are integers greater than 1, and w is a word of the form

κ > 1, 0 < a_i < l, 0 < β_i < m for ali i, which is not a proper power. We say that two words w and v are equivalent if we can transform one to the other by a sequence of the following moves

1. cyclic permutation;

2. inversion;

3. automorphism of ℤ_l or ℤ_m; or

4. interchanging the two free factors (if l = m);

and we write w ∼ ν. If in the presentation 1.1, we replace w by an equivalent word ν, then we get an isomorphic copy of G. Thus it is enough to study generalised triangle groups up to equivalence of w.

It is well known that the ordinary triangle groups

satisfy a Tits alternative. That is, they either contain a soluble subgroup of finite index or have a non-abelian free subgroup. In, Rosenberger asks whether a Tits alternative holds for generalised triangle groups.

Work in shows this conjecture to be true except possibly where κ > 4 and (l, m, n) = (3,3,2), (3,4,2), (3,5,2) or (2, m, 2) (m ≥ 3). It is this last case we address here.