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± 1) r ∈ 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 BS(1,k) , k≥ 2 , has a one-sided Følner sequence F such that the conjugacy ratio with respect to F is non-zero, even though BS(1,k) 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.