On a family of representations of residually finite groups

TL;DR

This paper constructs a family of unitary representations for residually finite groups based on a growth function, revealing different behaviors from regular to quasiregular representations and exploring intermediate cases.

Contribution

It introduces a new family of representations parameterized by growth functions, connecting them to known representations and analyzing their properties.

Findings

For minimal growth, the representation is weakly equivalent to the regular representation.

For maximal growth, it is weakly equivalent to a direct sum of quasiregular representations.

Intermediate growth cases exhibit diverse behaviors of the constructed representations.

Abstract

For a residually finite group $G$, its normal subgroups $G\supset G_1\supset G_2\cdots$ with $\cap_{n\in\mathbb N}G_n=\{e\}$ and for a growth function $\gamma$ we construct a unitary representation $\pi_\gamma$ of $G$. For the minimal growth, $\pi_\gamma$ is weakly equivalent to the regular representation, and for the maximal growth it is weakly equivalent to the direct sum of the quasiregular representations on the quotients $G/G_n$. In the case of intermediate growth we show two examples of different behaviour of $\pi_\gamma$.