Conjugacy problem in virtual right-angled Artin groups

TL;DR

This paper addresses the conjugacy problem in virtual right-angled Artin groups, establishing conditions for solvability, analyzing their geometric properties, and exploring the complexity of related algorithms.

Contribution

It introduces new results on the conjugacy problem for virtual RAAGs, including geometric criteria for solvability and analysis of the twisted conjugacy problem.

Findings

Virtual RAAGs with length-preserving automorphisms are CAT(0) and have solvable conjugacy problem.

The conjugacy growth series of these groups is transcendental.

The paper provides algorithms for the twisted conjugacy problem with complexity analysis.

Abstract

In this paper we solve the conjugacy problem for several classes of virtual right-angled Artin groups, using algebraic and geometric techniques. We show that virtual RAAGs of the form $A_{\phi} = A_{\Gamma} \rtimes_{\phi} \mathbb{Z}/m\mathbb{Z}$ are $\mathrm{CAT}(0)$ when $\phi \in \mathrm{Aut}(A_{\Gamma})$ is length-preserving, and so have solvable conjugacy problem. The geometry of these groups, namely the existence of contracting elements, allows us to show that the conjugacy growth series of these groups is transcendental. Examples of virtual RAAGs with decidable conjugacy problem for non-length preserving automorphisms are also studied. Finally, we solve the twisted conjugacy problem in RAAGs with respect to length-preserving automorphisms, and determine the complexity of this algorithm in certain cases.