Free group representations from vector-valued multiplicative functions. III

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