Coarse structures on groups defined by conjugations

TL;DR

This paper investigates the relationship between algebraic properties of groups and their coarse geometric structures, establishing conditions under which certain asymptotic dimensions and discreteness properties hold.

Contribution

It introduces a new coarse structure on groups based on conjugation and characterizes when the asymptotic dimension is zero or when the subgroup coarse space is discrete.

Findings

asdim of G is zero iff G/Z_G is locally finite

coarse space of subgroups is discrete iff G is Dedekind

connects algebraic properties with coarse geometric features

Abstract

For a group $G$, we denote by $\stackrel{\leftrightarrow}{G}$ 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 $\stackrel{\leftrightarrow}{G}$. In particular, we show that $asdim \ \stackrel{\leftrightarrow}{G} = 0$ 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.