On conjugacy separability of graph products of groups

TL;DR

This paper proves that certain classes of conjugacy separable groups, including right angled Coxeter and Artin groups, are preserved under graph products, expanding understanding of their algebraic structure.

Contribution

It establishes that $ ext{C}$-hereditarily conjugacy separable groups are closed under graph products for extension-closed classes $ ext{C}$, and applies this to specific group classes.

Findings

Right angled Coxeter groups are hereditarily conjugacy separable.

Infinitely generated right angled Artin groups are hereditarily conjugacy separable.

The class of $ ext{C}$-conjugacy separable groups is closed under graph products.

Abstract

We show that the class of $\mathcal{C}$-hereditarily conjugacy separable groups is closed under taking arbitrary graph products whenever the class $\mathcal{C}$ is an extension closed variety of finite groups. As a consequence we show that the class of $\mathcal{C}$-conjugacy separable groups is closed under taking arbitrary graph products. In particular, we show that right angled Coxeter groups are hereditarily conjugacy separable and 2-hereditarily conjugacy separable, and we show that infinitely generated right angled Artin groups are hereditarily conjugacy separable and $p$-hereditarily conjugacy separable for every prime number $p$.