Free Group Representations from Vector-Valued Multiplicative Functions, II

TL;DR

This paper studies multiplicative representations of free groups, providing criteria for their irreducibility based on matrix coefficient growth, extending previous work on their structure and properties.

Contribution

It offers a new irreducibility criterion for multiplicative representations of free groups, analyzing their behavior as group representations.

Findings

Established a growth-based criterion for irreducibility.

Extended previous constructions of multiplicative representations.

Analyzed the representations' properties as group representations.

Abstract

Let $\Gamma$ be a non-commutative free group on finitely many generators. In a previous work two of the authors have constructed the class of multiplicative representations of $\Gamma$ and proved them irreducible as representation of $\Gamma\ltimes_\lambda C(\Omega)$. In this paper we analyze multiplicative representations as representations of $\Gamma$ and we prove a criterium for irreducibility based on the growth of their matrix coefficients.