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

## TL;DR

This paper studies the structure of certain subgroups of countable groups via quasi-regular representations and explores their impact on associated C*-algebras, revealing rigidity and ideal structure results.

## Contribution

It introduces new equivalence relations on subgroups based on quasi-regular representations and identifies classes with rigidity properties and their influence on C*-algebra structures.

## Key findings

- Subgroups with spectral gap are rigid under various equivalences.
- The ideal structure of C*-algebras is characterized for subgroups with spectral gap or weakly parabolic properties.
- Results extend to induced representations beyond quasi-regular cases.

## Abstract

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 $H\in {\rm Sub}(G)$.