The Conjugacy Ratio of Abelian-by-Cyclic Groups

Abstract

Let $G=K⋊⟨t⟩$ be a finitely generated group where $K$ is abelian and $⟨t⟩$ 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 Følner 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 Følner sequences, where Tointon showed that the conjugacy ratio with respect to a two-sided Følner sequence is positive if and only if the group is virtually abelian.

arXiv comment19 pages. Accepted for publication in Proceedings of the Edinburgh Mathematical Society

References in corpus (4)

• Commuting probabilities of infinite groups
• The rationality of Sol manifolds
• The conjugacy growth of the soluble Baumslag-Solitar groups
• The conjugacy ratio of groups