The dynamical approach to the conjugacy in groups

TL;DR

This paper explores the relationship between algebraic properties of a discrete group and the dynamical behavior of its action on the Stone-Čech compactification, revealing conditions under which conjugacy classes are finite.

Contribution

It introduces a dynamical framework for understanding conjugacy in groups via ultrafilters and characterizes groups with finite conjugacy classes using this approach.

Findings

Finite conjugacy classes correspond to finite orbits in the ultrafilter space.

The commutant of the group is finite if and only if all ultrafilter conjugacy classes are finite.

Abstract

Given a discrete group $G$, we identify the Stone-$\check C$ech compactification $\beta G$ with the set of all ultrafilters on $G$ and put $G^\ast =\beta G\setminus G$. The action $G$ on $G$ by the conjugations $(g,x)\mapsto g^{-1}xg$ induces the action of $G$ on $G^\ast$ by $(g, p)\mapsto p^g $, $p^g = \{ g^{-1} Pg: P\in p\}$. We study interplays between the algebraic properties of $G$ and the dynamical properties of $(G, G^\ast)$. In particular, we show that $p^G$ is finite for each $p\in G^\ast$ if and only if the commutant of $G$ is finite.