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

Conjugacy languages in virtual graph products