Virtually compact special hyperbolic groups are conjugacy separable

TL;DR

This paper proves that virtually compact special hyperbolic groups are conjugacy separable, extending this property to many Coxeter and small cancellation groups using a new criterion involving discrete and profinite (co)homology.

Contribution

It introduces a novel criterion for conjugacy separability of prime order elements and applies it to a broad class of hyperbolic groups, advancing understanding in geometric group theory.

Findings

Virtually compact special hyperbolic groups are conjugacy separable.

Many hyperbolic Coxeter and small cancellation groups are conjugacy separable.

A new criterion for conjugacy distinguished elements of prime order is established.

Abstract

We prove that any word hyperbolic group which is virtually compact special (in the sense of Haglund and Wise) is conjugacy separable. As a consequence we deduce that all word hyperbolic Coxeter groups and many classical small cancellation groups are conjugacy separable. To get the main result we establish a new criterion for showing that elements of prime order are conjugacy distinguished. This criterion is of independent interest; its proof is based on a combination of discrete and profinite (co)homology theories.