Arbitrary residual finiteness and conjugacy separability growth

TL;DR

This paper demonstrates that for any sufficiently fast growing functions, there exist groups with matching residual finiteness and conjugacy separability growth, and it constructs groups where these two growth functions can vary independently.

Contribution

It extends Bradford's results by constructing groups with prescribed residual finiteness and conjugacy separability growth functions, showing their independent variability.

Findings

Groups can have arbitrary fast growing residual finiteness functions.

Conjugacy separability growth can match residual finiteness growth.

Residual finiteness and conjugacy separability growth functions can be arbitrarily far apart.

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 $f_2\geq f_1$, 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.