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

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Let G be a countable group. We introduce several equivalence relations on the set Sub(G) of subgroups of G, defined by properties of the quasi-regular representations λ G/H associated to H ∈ Sub(G) and compare them to the relation of G-conjugacy of subgroups. We define a class Sub 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 ∈ Sub sg (G) for any one of the above equivalence relations coincides with the G-conjugacy class of H. Next, we introduce a second class Sub 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