On the conjugacy growth functions of groups

TL;DR

This paper investigates the relationship between conjugacy growth functions and ordinary growth functions in finitely generated groups, proposing that exponential growth in one often implies exponential growth in the other, supported by conjectures and examples.

Contribution

It introduces conjectures and examples suggesting that groups with exponential ordinary growth typically also have exponential conjugacy growth, clarifying their relationship.

Findings

Conjugacy growth can be constant despite exponential ordinary growth.

Groups with exponential growth often have exponential conjugacy growth.

The paper presents conjectures and examples supporting this correlation.

Abstract

To every finitely generated group one can assign the conjugacy growth function that counts the number of conjugacy classes intersecting a ball of radius $n$. Results of Ivanov and Osin show that the conjugacy growth function may be constant even if the (ordinary) growth function is exponential. The aim of this paper is to provide conjectures, examples and statements that show that in "normal" cases, groups with exponential growth functions also have exponential conjugacy growth functions.