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

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').