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Conjugacy languages in virtual graph products

Gemma Crowe

TL;DR

This work advances the understanding of language-theoretic properties for conjugacy in virtual graph products, focusing on semi-direct extensions $G_{\rm φ}=G_{Γ}\rtimes_{\phi}\langle t\rangle$ with finite-order automorphisms. It establishes conditions under which the conjugacy-geodesic language $\mathsf{ConjGeo}$ is regular, and it identifies criteria that yield non-context-free or non-unambiguous-context-free behavior for the spherical conjugacy language $\mathsf{ConjSL}$, including in virtual RAAGs. The authors develop a robust twisted-conjugacy framework in RAAGs, proving a twisted cyclic-reduction theorem for $\psi$-conjugacy and analyzing the impact of inversions and other automorphisms on language regularity. They extend these results to conjugacy shortlex languages in virtual graph products, connecting AH-accessibility and acylindrical hyperbolicity to complexity classes, and provide a suite of transfer principles that propagate (non-)CF properties along induced subgraphs and extensions. Overall, the paper broadens the landscape of how conjugacy-growth-related languages behave in graph-product-like groups and offers tools for assessing algorithmic and growth-theoretic aspects in these families.

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.

Conjugacy languages in virtual graph products

TL;DR

This work advances the understanding of language-theoretic properties for conjugacy in virtual graph products, focusing on semi-direct extensions with finite-order automorphisms. It establishes conditions under which the conjugacy-geodesic language is regular, and it identifies criteria that yield non-context-free or non-unambiguous-context-free behavior for the spherical conjugacy language , including in virtual RAAGs. The authors develop a robust twisted-conjugacy framework in RAAGs, proving a twisted cyclic-reduction theorem for -conjugacy and analyzing the impact of inversions and other automorphisms on language regularity. They extend these results to conjugacy shortlex languages in virtual graph products, connecting AH-accessibility and acylindrical hyperbolicity to complexity classes, and provide a suite of transfer principles that propagate (non-)CF properties along induced subgraphs and extensions. Overall, the paper broadens the landscape of how conjugacy-growth-related languages behave in graph-product-like groups and offers tools for assessing algorithmic and growth-theoretic aspects in these families.

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.
Paper Structure (19 sections, 52 theorems, 106 equations, 2 figures)

This paper contains 19 sections, 52 theorems, 106 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group extensions
  4. Twisted conjugacy
  5. Graph products
  6. Automorphisms of graph products
  7. Languages and growth series
  8. Languages in groups
  9. Twisted conjugacy languages
  10. Twisted conjugacy in RAAGs
  11. Twisted cyclic reduction: inversions
  12. Conjugacy geodesics in virtual graph products
  13. Non-length preserving automorphisms and conjugacy geodesics
  14. Graph products
  15. The Conjugacy shortlex language in virtual graph products
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $A_{\Gamma} = \langle X \rangle$ be a RAAG, and let $\psi \in \mathrm{Aut}(A_{\Gamma})$. Let $u,v \in X^{\ast}$ be two $\psi$-cyclically reduced words. Then $u$ and $v$ are $\psi$-conjugate if and only if $u$ and $v$ are related by a finite sequence of $\psi$-cyclic permutations, commutation rel

Figures (2)

  • Figure 1: Automata $M_{i,k}(q,q')$
  • Figure 2: Partial picture of automaton $M'$ which accepts $\mathsf{ConjGeo}\left(G, \widehat{X}\right)$

Theorems & Definitions (107)

  • Theorem 1.1
  • Theorem 1.1
  • Corollary 1.0
  • Corollary 1.0
  • Corollary 1.0
  • Corollary 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.3
  • Lemma 2.1
  • ...and 97 more