Conjugacy languages and conjugacy growth relative to subsets of groups

TL;DR

This paper investigates conjugacy languages in groups with constraints, analyzing their regularity and properties relative to subsets, and introduces a relative conjugacy growth concept showing dependence on subset choice.

Contribution

It extends the study of conjugacy languages to constrained problems and various group classes, providing new regularity results and a framework for relative conjugacy growth.

Findings

In free groups, certain conjugacy languages are regular for rational subsets.

In hyperbolic groups, regularity of conjugacy languages depends on geodesic languages.

In virtually cyclic and abelian groups, conjugacy languages exhibit specific regularity properties.

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.