The conjugacy ratio of groups

TL;DR

This paper introduces the conjugacy ratio for finitely generated groups, conjectures it is zero except for virtually abelian groups, and confirms this for several classes including hyperbolic and lamplighter groups.

Contribution

It defines a new invariant called the conjugacy ratio and provides evidence supporting the conjecture that it vanishes for most groups beyond virtually abelian ones.

Findings

Conjecture that the conjugacy ratio is zero for all non-virtually abelian groups.

Confirmed the conjecture for residually finite groups of subexponential growth.

Confirmed the conjecture for hyperbolic, right-angled Artin, and lamplighter groups.

Abstract

In this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is $0$ for all groups except the virtually abelian ones, and confirm this conjecture for certain residually finite groups of subexponential growth, hyperbolic groups, right-angled Artin groups, and the lamplighter group.