Quantifying conjugacy separability in wreath products of groups

We study generalisations of conjugacy separability in restricted wreath products of groups. We provide an effective upper bound for $\mathcal{C}$-conjugacy separability of a wreath product $A \wr B$ in terms of the $\mathcal{C}$-conjugacy separability of $A$ and $B$, the growth of $\mathcal{C}$-cyclic subgroup separability of $B$, and the $\mathcal{C}$-residual girth of $B.$ As an application, we provide a characterisation of when $A \wr B$ is $p$-conjugacy separable. We use this characterisation to the provide for each prime $p$ an example of wreath products with infinite base group that are $p$-conjugacy separable. We also provide asymptotic upper bounds for conjugacy separability for wreath products of nilpotent groups which include the lamplighter groups and provide asymptotic upper bounds for conjugacy separability of the free metabelian groups. Along the way, we provide a polynomial upper bound for the shortest conjugator between two elements of length at most $n$ in a finitely generated nilpotent group.