Conjugacy languages and conjugacy growth relative to subsets of groups

TL;DR

This work extends the language-theoretic analysis of conjugacy to the generalized setting with subset constraints. It defines and studies relative conjugacy languages $\text{ConjGeo}(U,V)$, $\text{CycGeo}(U)$, $\text{ConjSL}(U)$ and $\text{ConjMinLenSL}(U,V)$ across several group classes, proving regularity results for free groups, hyperbolic groups (under quasiconvexity), virtually cyclic groups, and certain virtually abelian configurations. A key contribution is the exploration of relative conjugacy growth, showing that in free groups relative growth can be polynomial of any degree (and that only polynomial or exponential growth occurs overall), with constructive examples $U_d$ producing degree $d-1$ growth. The paper also links these language-theoretic properties to decidability: under suitable closure conditions, regularity of relative conjugacy languages yields decidability of the $\mathcal{C}$-generalized conjugacy problem with $\mathcal{C}$-constraints. These results unify and extend prior findings, offering a framework for analyzing generalized conjugacy in broader group-theoretic and formal-language contexts.

Abstract

In this paper, we explore conjugacy languages when the base problem is the generalized conjugacy problem (with constraints): given $g\in G$ and $U\subset G$, does $g$ have a conjugate in $U$ (with conjugators in a certain subset)? To do so, for subsets $U,V\subseteq G$, we define the corresponding languages $\text{ConjGeo(U,V)}$, $\text{CycGeo(U)}$, $\text{ConjSL(U)}$ and $\text{ConjMinLenSL(U,V)}$, following the previously studied cases where $U=V=G$. Our results cover several classes of groups: for free groups, we prove that $\text{ConjGeo(U,V)}$ and $\text{ConjMinLenSL(U,V)}$ are regular if $U$ and $V$ are rational subsets; for hyperbolic groups, we show that if $L$ is a regular language of geodesics and $U$ is the subsets represented by it, then $\text{ConjGeo(U)}$ and $\text{ConjMinLenSL(U)}$ are regular; for virtually cyclic groups, we show that $\text{ConjSL(U)}$ is regular if $U$ is rational; and, for virtually abelian groups, we prove that $\text{ConjGeo(U)}$ belongs to a certain class of languages $\C$ when the language of words representing elements of $U$ also belongs to $\C$. We also define relative conjugacy growth and show that its behavior can be heavily dependent on the choice of subset.

Theorems & Definitions (53)

• Theorem 1.1
• Proposition 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5: Benois, [Ben79]
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8
• Theorem 2.9