Coarse structures on groups defined by conjugations

For a group $G$, we denote by $\stackrel{\leftrightarrow}{G}$ the coarse space on $G$ endowed with the coarse structure with the base $\{\{(x,y)\in G\times G\colon y\in x^F \} \colon F \in [G]^{<\omega} \}$, $x^F = \{z^{-1} xz\colon z\in F \}$. Our goal is to explore interplays between algebraic properties of $G$ and asymptotic properties of $\stackrel{\leftrightarrow}{G}$. In particular, we show that $\operatorname{asdim}\stackrel{\leftrightarrow}{G} = 0$ 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.