The Conjugacy Ratio of Abelian-by-Cyclic Groups

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 $\mathrm{B}\mathrm{S}(1,2)$ has a one-sided Følner sequence $F$ such that the conjugacy ratio with respect to $F$ is non-zero, even though $\mathrm{B}\mathrm{S}(1,2)$ 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.