Let $G$ be a countable group. We introduce several equivalence relations on the set ${ \rm Sub}(G)$ of subgroups of $G$, defined by properties of the quasi-regular representations $\lambda_{ G/H}$ associated to $H\in { \rm Sub}(G)$ and compare them to the relation of $G$-conjugacy of subgroups. We define a class ${ \rm Sub}_{ \rm sg}(G)$ 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 $H\in { \rm Sub}_{ \rm sg}(G)$ for any one of the above equivalence relations coincides with the $G$-conjugacy class of $H$. Next, we introduce a second class ${ \rm Sub}_{ \rm w-par}(G)$ 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 ${ \rm Sub}_{ \rm w-par}(G)$ and ${ \rm Sub}_{ \rm sg}(G)$. Our results are valid, more generally, for induced representations ${ \rm Ind}_H^G \sigma$, where $\sigma$ is a representation of $H\in { \rm Sub}(G)$.
Categories: math.GR math.OA