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

Let $G$ be a countable group. We introduce several equivalence relations on the set $\mathrm{S}\mathrm{u}\mathrm{b}(G)$ of subgroups of $G$, defined by properties of the quasi-regular representations ${\lambda }_{G\mathrm{/}H}$ associated to $H\in \mathrm{S}\mathrm{u}\mathrm{b}(G)$ and compare them to the relation of $G$-conjugacy of subgroups. We define a class ${\mathrm{S}\mathrm{u}\mathrm{b}}_{\mathrm{s}\mathrm{g}}(G)$ 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 $H\in {\mathrm{S}\mathrm{u}\mathrm{b}}_{\mathrm{s}\mathrm{g}}(G)$ for any one of the above equivalence relations coincides with the $G$-conjugacy class of $H$. Next, we introduce a second class ${\mathrm{S}\mathrm{u}\mathrm{b}}_{\mathrm{w}-\mathrm{p}\mathrm{a}\mathrm{r}}(G)$ of subgroups (these are subgroups which are weakly parabolic in some sense) and we establish results concerning the ideal structure of the $C^{\ast }$-algebra $C_{{\lambda }_{G\mathrm{/}H}}^{\ast }(G)$ generated by ${\lambda }_{G\mathrm{/}H}$ for subgroups $H$ which belong to either one of the classes ${\mathrm{S}\mathrm{u}\mathrm{b}}_{\mathrm{w}-\mathrm{p}\mathrm{a}\mathrm{r}}(G)$ and ${\mathrm{S}\mathrm{u}\mathrm{b}}_{\mathrm{s}\mathrm{g}}(G)$. Our results are valid, more generally, for induced representations ${\mathrm{I}\mathrm{n}\mathrm{d}}_H^G\sigma$, where $\sigma$ is a representation of $H\in \mathrm{S}\mathrm{u}\mathrm{b}(G)$.