Arbitrary residual finiteness and conjugacy separability growth

Abstract

In a recent paper, Henry Bradford showed that all sufficiently fast growing functions appear as the residual finiteness growth function of some group. In this paper we show that the groups there constructed are conjugacy separable and that their conjugacy separability growth is equal to the residual finiteness growth. It follows that all sufficiently fast growing functions appear as the conjugacy separability growth function of some group. We extend this construction to a new class of groups such that given functions $f_1,f_2$ under the same constraints and satisfying $f_2\ge f_1$, we can find a group such that the residual finiteness growth is given by $f_1$ and the conjugacy separability growth by $f_2$, showing that the residual finiteness growth and conjugacy separability growth behave independently and can lie arbitrarily far apart.