Conjugacy languages in virtual graph products

Gemma Crowe

doi:10.1016/j.jalgebra.2023.06.029

Conjugacy languages in virtual graph products

Gemma Crowe

School of Mathematical & Computer Sciences

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Abstract

In this paper we study the behaviour of conjugacy languages in virtual graph products, extending results by Ciobanu et al. [13]. 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.

Original language	English
Pages (from-to)	873-910
Number of pages	38
Journal	Journal of Algebra
Volume	634
Early online date	13 Jul 2023
DOIs	https://doi.org/10.1016/j.jalgebra.2023.06.029
Publication status	Published - 15 Nov 2023

Keywords

Conjugacy growth
Formal languages
Right-angled Artin groups
Twisted conjugacy

ASJC Scopus subject areas

Algebra and Number Theory

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abstract = "In this paper we study the behaviour of conjugacy languages in virtual graph products, extending results by Ciobanu et al. [13]. 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.",

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AB - In this paper we study the behaviour of conjugacy languages in virtual graph products, extending results by Ciobanu et al. [13]. 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.

KW - Conjugacy growth

KW - Formal languages

KW - Right-angled Artin groups

KW - Twisted conjugacy

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Conjugacy languages in virtual graph products

Abstract

Keywords

ASJC Scopus subject areas

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