Quasi-regular representations of discrete groups and associated $C^*$-algebras

Abstract:

Let $G$ be a countable group. We introduce several equivalence relations on the set $\operatorname {Sub}(G)$ of subgroups of $G$, defined by properties of the quasi-regular representation $\lambda _{G/H}$ associated with $H\in \operatorname {Sub}(G)$, and we compare them to the relation of $G$-conjugacy of subgroups. We define a class $\operatorname {Sub}_{\mathrm {sg}}(G)$ of subgroups (these are subgroups with a certain spectral gap property) and show that they are rigid, in the sense that the equivalence class of $H\in \operatorname {Sub}_{\mathrm {sg}}(G)$ for any one of the above equivalence relations coincides with the $G$-conjugacy class of $H$. Next, we introduce a second class $\operatorname {Sub}_{\text {w-par}}(G)$ of subgroups (these are subgroups which are weakly parabolic in some sense), and we establish results concerning the ideal structure of the $C^*$-algebra $C^*_{\lambda _{G/H}}(G)$ generated by $\lambda _{G/H}$ for subgroups $H$ which belong to either one of the classes $\operatorname {Sub}_{\text {w-par}}(G)$ and $\operatorname {Sub}_{\mathrm {sg}} (G)$. Our results are valid, more generally, for induced representations $\operatorname {Ind}_H^G \sigma$, where $\sigma$ is a representation of $H\in \operatorname {Sub}(G)$.