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.