Coarse structures on groups defined by conjugations

Abstract

For a group $G$, we denote by $\overleftrightarrow{G}$ the coarse space on $G$ endowed with the coarse structure with the base $\{\{(x,y)\in G\times G:y\in x^F\}:F\in [G{]}^{<\omega }\}$, $x^F=\{z^{-1}xz:z\in F\}$. Our goal is to explore interplays between algebraic properties of $G$ and asymptotic properties of $\overleftrightarrow{G}$. In particular, we show that $asdim\overleftrightarrow{G}=0$ if and only if $G/Z_G$ is locally finite, $Z_G$ is the center of $G$. For an infinite group $G$, the coarse space of subgroups of $G$ is discrete if and only if $G$ is a Dedekind group.

coarse structure defined by conjugations, cellularity, FC-group, ultrafilter