Effective conjugacy separability of virtually abelian groups

Abstract

A natural question for groups $H$ is which data can be detected in its finite quotients. A subset $X \subset H$ is called separable if for all $h\in H \setminus X$, there exists an epimorphism $\varphi$ to a finite group $Q$ such that $\varphi(h)\notin\varphi(X)$. More specifically, a group is said to be conjugacy separable if every conjugacy class is separable. It is known that many classes of groups are conjugacy separable, including virtually free and polycyclic groups. The minimal order of the quotient $Q$, in terms of the complexity of the conjugacy classes under consideration, is captured by the conjugacy separability function $\mathrm{Conj}_H: \mathbb{N} \to \mathbb{N}$. This function is in general ill understood, in fact the only large class of groups for which it is known exactly are the abelian groups. Indeed, in this case $\mathrm{Conj}_H$ is equal to the residual finiteness function, that is the size of quotients needed to separate singletons, and thus logarithmic if the group is infinite. Recent work has described the residual finiteness function for the class of virtually abelian groups, which gives a lower bound for the conjugacy separability function. The main result of this paper is a characterization of $\mathrm{Conj}_H$ for every virtually abelian group $H$. If the corresponding extension is associated with an irreducible representation over $\mathbb{Q}$, we demonstrate that we obtain the same function as the residual finiteness function. However, if the representation is not irreducible, we find an expression that is in some cases strictly larger, which we illustrate with several examples.