Two new proofs of the fact that triangle groups are distinguished by their finite quotients

Marston Conder

doi:10.53733/193

Abstract

In a 2016 paper by Alan Reid, Martin Bridson and the author, it was shown using the theory of profinite groups that if $\Gamma$ is a finitely-generated Fuchsian group and $\Sigma$ is a lattice in a connected Lie group, such that $\Gamma$ and $\Sigma$ have exactly the same finite quotients, then $\Gamma$ is isomorphic to $\Sigma$. As a consequence, two triangle groups $\Delta(r,s,t)$ and $\Delta(u,v,w)$ have the same finite quotients if and only if $(u,v,w)$ is a permutation of $(r,s,t)$. A direct proof of this property of triangle groups was given in the final section of that paper, with the purpose of exhibiting explicit finite quotients that can distinguish one triangle group from another. Unfortunately, part of the latter direct proof was flawed. In this paper two new direct proofs are given, one being a corrected version using the same approach as before (involving direct products of small quotients), and the other being a shorter one that uses the same preliminary observations as in the earlier version but then takes a different direction (involving further use of the `Macbeath trick').

Highlights

For any positive integers r, s and t, the ordinary (r, s, t) triangle group is the abstract group ∆(r, s, t) with presentation x, y, z | xr = ys = zt = xyz = 1 .In a recent paper [2], it was shown that if Γ is a finitely-generated Fuchsian group and Σ is a lattice in a connected Lie group, such that Γ and Σ have exactly the same finite quotients, Γ is isomorphic to Σ
A second proof of the fact concerning triangle groups was given in the final section of [2], with the purpose of exhibiting explicit finite quotients that can distinguish one triangle group from another
The use of direct products of two quotients each isomorphic to PSL(2, p) for some prime p did not cover all possibilities remaining after simpler methods were applied

Summary

Introduction

A second proof of the fact concerning triangle groups was given in the final section of [2], with the purpose of exhibiting explicit finite quotients that can distinguish one triangle group from another. The second proof is quite a lot shorter, using certain smooth quotients of triangle groups ∆(k, l, m), where k, l and m divide either r, s and t, or u, v and w (in some order) This is an approach we considered earlier and were able to use successfully to deal with the vast majority of triple pairs, but completed only with the help of a key observation made by Frankie Chan in [3] for the remaining pairs, and again we owe a debt of gratitude to him for that. Before giving the two new proofs, we repeat and extend some of the background to this topic, in order to make the paper self-contained

Further Background

Main Theorem and Preliminary Steps for its New Proofs