Profinite properties of Coxeter groups

TL;DR

This paper investigates the profinite completions of Coxeter groups, establishing their properties, rigidity results, and invariance of certain algebraic invariants, thereby advancing understanding of their algebraic and geometric structures.

Contribution

It proves Coxeter groups are good in Serre's sense, detects splittings via profinite completions, and establishes various forms of profinite rigidity for classes of Coxeter groups.

Findings

Coxeter groups are good in the sense of Serre.

Splittings of Coxeter groups are detectable through profinite completions.

Certain Coxeter groups are profinitely rigid among Coxeter groups.

Abstract

We prove a number of results about profinite completions of Coxeter groups. For example we prove Coxeter groups are good in the sense of Serre and that various splittings of Coxeter groups arising from actions on trees are detected by the profinite completion. As an application we prove a number of families Coxeter groups are profinitely rigid amongst Coxeter groups. We also prove that Gromov-hyperbolic FC type, large type, and odd Coxeter groups are almost profinitely rigid amongst Coxeter groups. In the appendix, Sam Fisher and Sam Hughes show that the Atiyah Conjecture holds for all Coxeter groups, and that $\ell^2$-Betti numbers and their positive characteristic analogues are profinite invariants of Coxeter groups and of virtually compact special groups.