Conjugacy problem in virtual right-angled Artin groups

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.