The Conjugacy Ratio of Abelian-by-Cyclic Groups

TL;DR

This paper investigates the conjugacy ratio in abelian-by-cyclic groups, proving it is zero unless the group is virtually abelian, and explores contrasting behaviors in Baumslag--Solitar groups with different F{ exto}lner sequences.

Contribution

It confirms a conjecture that the conjugacy ratio is zero for non-virtually abelian abelian-by-cyclic groups and shows non-zero conjugacy ratio in Baumslag--Solitar groups with specific F{ exto}lner sequences.

Findings

Conjugacy ratio is zero unless the group is virtually abelian.

Baumslag--Solitar group BS(1,2) has a non-zero conjugacy ratio with a one-sided F{ exto}lner sequence.

Two-sided F{ exto}lner sequences yield positive conjugacy ratio only for virtually abelian groups.

Abstract

Let $G = K \rtimes \langle t \rangle $ be a finitely generated group where $K$ is abelian and $\langle t\rangle$ is the infinite cyclic group. Let $ R $ be a finite symmetric subset of $K$ such that $S = \{ (r,1),(0,t^{\pm 1}) \mid r \in R \}$ is a generating set of $G$. We prove that the spherical conjugacy ratio, and hence the conjugacy ratio, of $G$ with respect to $S$ is $0$ unless $G$ is virtually abelian, confirming a conjecture of Ciobanu, Cox and Martino in this case. We also show that the Baumslag--Solitar group $\mathrm{BS}(1,2)$ has a one-sided F{\o}lner sequence $F$ such that the conjugacy ratio with respect to $F$ is non-zero, even though $\mathrm{BS}(1,2)$ is not virtually abelian. This is in contrast to two-sided F{\o}lner sequences, where Tointon showed that the conjugacy ratio with respect to a two-sided F{\o}lner sequence is positive if and only if the group is virtually abelian.