Conjugacy languages in virtual graph products

TL;DR

This paper investigates the properties of conjugacy languages in virtual graph products, especially focusing on regularity and context-freeness, extending prior results to new classes of groups.

Contribution

It extends the analysis of conjugacy languages to virtual graph products, providing new regularity results and criteria for non-context-freeness.

Findings

Regularity of conjugacy geodesic language in certain virtual graph products

Properties of spherical conjugacy language depend on automorphism and ordering

Spherical conjugacy language is not unambiguous context-free in some cases

Abstract

In this paper we study the behaviour of conjugacy languages in virtual graph products, extending results by Ciobanu, Hermiller, Holt and Rees. We focus primarily on virtual graph products in the form of a semi-direct product. First, we study the behaviour of twisted conjugacy representatives in right-angled Artin and Coxeter groups. We prove regularity of the conjugacy geodesic language for virtual graph products in certain cases, and highlight properties of the spherical conjugacy language, depending on the automorphism and ordering on the generating set. Finally, we give a criterion for when the spherical conjugacy language is not unambiguous context-free for virtual graph products. We can extend this further in the case of virtual RAAGs, to show the spherical conjugacy language is not context-free.