The regular and profinite representations of residually finite groups

TL;DR

This paper explores conditions under which certain unitary representations of residually finite groups, constructed from decreasing sequences of finite index subgroups, weakly contain the regular representation, shedding light on their structural properties.

Contribution

It characterizes when the constructed representations from subgroup sequences weakly contain the regular representation in residually finite groups.

Findings

Provides criteria for weak containment of the regular representation

Analyzes the relationship between subgroup sequences and representation properties

Enhances understanding of the structure of residually finite groups

Abstract

Let $G$ be a residually finite group. To any decreasing sequence $\mathcal S = (H_n)_n $ of finite index subgroups of $G$ is associated a unitary representation $\rho_{\mathcal S}$ of $G$ in the Hilbert space $\bigoplus_{n=0}^{+\infty} \ell^2 (G/H_n) $. This paper investigates the following question: when does the representation $\rho_{\mathcal S} $ weakly contain the regular representation $\lambda$ of $G$?