The regular and profinite representations 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 ${⨁}_{n=0}^{+\infty }{\ell }^2(G/H_n)$. This paper investigates the following question: when does the representation ${\rho }_{\mathcal{S}}$ weakly contain the regular representation $\lambda$ of $G$?

19 pages, 1 figure