Twisted commutativity and conjugacy ratio in groups

TL;DR

This paper introduces and analyzes the concepts of twisted commutativity degree and twisted conjugacy ratio in finitely generated groups, providing explicit computations for various classes and exploring their properties and implications.

Contribution

It defines new measures of twisted commutativity and conjugacy ratios, computes these for several classes of groups, and investigates their behavior and significance in group theory.

Findings

Computed twisted commutativity degree for virtually abelian, subexponential, and free groups.

Calculated twisted conjugacy ratio for virtually abelian groups.

Provided examples of groups with exponential growth where the ratio is zero.

Abstract

In this paper we introduce and study the degree of twisted commutativity and the twisted conjugacy ratio of a finitely generated group $G$. The degree of twisted commutativity $\mathrm{tdc}_X(\varphi, G)$ generalises the degree of commutativity of $G$, by measuring the density of pairs of elements with trivial twisted commutators in the ball of radius $n$ of $G$, as $n \rightarrow \infty$, where the twisting is done with respect to an endomorphism $\varphi$ of $G$. We compute $\mathrm{tdc}_X(\varphi, G)$ for several classes of groups, including virtually abelian groups, groups of subexponential growth, and free groups. We then study the twisted conjugacy ratio $\mathrm{tcr}_{X}(\varphi, G)$, which is the limit at infinity of the quotient of the twisted conjugacy and standard growth functions. We compute $\mathrm{tcr}_{X}(\varphi, G)$ for virtually abelian groups, and give examples of groups of exponential growth such that $\mathrm{tcr}_{X}(\varphi, G) = 0$.