Hereditary conjugacy separability of right angled Artin groups and its applications

TL;DR

This paper proves that finite index subgroups of right angled Artin groups are conjugacy separable and applies this to show similar properties in other groups, impacting their automorphism groups and decision problems.

Contribution

It establishes hereditary conjugacy separability for right angled Artin groups and extends this property to hyperbolic Coxeter groups and Bestvina-Brady groups.

Findings

Finite index subgroups of right angled Artin groups are conjugacy separable.

Word hyperbolic Coxeter groups contain conjugacy separable subgroups of finite index.

Bestvina-Brady groups are conjugacy separable and have solvable conjugacy problem.

Abstract

We prove that finite index subgroups of right angled Artin groups are conjugacy separable. We then apply this result to establish various properties of other classes of groups. In particular, we show that any word hyperbolic Coxeter group contains a conjugacy separable subgroup of finite index and has a residually finite outer automorphism group. Another consequence of the main result is that Bestvina-Brady groups are conjugacy separable and have solvable conjugacy problem.