Stability properties of multiplicative representations of the free group

In 2004 the second and third authors introduced a large family of representations of a free group Gamma weakly contained in the regular representation.In this paper we enlarge a little bit this class for Gamma so that the new class Mult(Gamma), of the multiplicative representations, is stable under taking finite direct sums, under restriction to, and induction from a finite index subgroup.As an application, using the properties of multiplicative representations we define a new class of tempered unitary representations for a class of groups that includes for example all lattices of unimodular subgroups of automorphisms of a locally finite regular tree.The main tool is the detailed study of the properties of the action of a free group on its Cayley graph with respect to a change of generators, as well as the relative properties of the action of a subgroup of finite index after the choice of a nice fundamental domain.