Free group representations from vector-valued multiplicative functions, III

TL;DR

This paper proves a conjecture about the classification of certain irreducible unitary representations of free groups, showing they are either odd, monotonous, or duplicitous, by analyzing multiplicative representations derived from the group's Cayley graph.

Contribution

It establishes the conjecture for all multiplicative representations of free groups, completing the classification of these representations.

Findings

All multiplicative representations are either odd, monotonous, or duplicitous.

The conjecture holds for the class of multiplicative representations.

The classification confirms the conjectured structure of these representations.

Abstract

Let $\pi$ be an irreducible unitary representation of a finitely generated nonabelian free group $\Gamma$; suppose $\pi$ is weakly contained in the regular representation. In 2001 the first and third authors conjectured that such a representation must be either odd or monotonous or duplicitous. In 2004 they introduced the class of multiplicative representations: this is a large class of representations obtained by looking at the action of $\Gamma$ on its Cayley graph. In the second paper of this series we showed that some of the multiplicative representations were monotonous. Here we show that all the other multiplicative representations are either odd or duplicitous. The conjecture is therefore established for multiplicative representations.