Higman-Thompson groups and profinite properties of right-angled Coxeter groups

TL;DR

This paper proves that right-angled Coxeter groups are uniquely determined by their profinite completions within Coxeter groups, explores their profinite genus, and extends rigidity results to graph products, also analyzing Higman-Thompson groups.

Contribution

It establishes profinite rigidity for RACGs within Coxeter groups and extends rigidity results to graph products, also showing Higman-Thompson groups are generated by four involutions.

Findings

RACGs are profinitely rigid among Coxeter groups.

Some RACGs have infinite profinite genus among residually finite groups.

Higman-Thompson groups are generated by four involutions.

Abstract

We prove that every right-angled Coxeter group (RACG) is profinitely rigid amongst all Coxeter groups. On the other hand we exhibit RACGs which have infinite profinite genus amongst all finitely generated residually finite groups. We also establish profinite rigidity results for graph products of finite groups. Along the way we prove that the Higman-Thompson groups $V_{n}$ are generated by $4$ involutions, generalising a classical result of Higman for Thompson's group $V$.