The dynamical approach to the conjugacy in groups

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.

space $G^{\ast }$ of ultrafilters on a group $G$, the action of $G$ on $G^{\ast }$ by the conjugations, ultracenter of $G^{\ast }$