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 $\mathbf {Mult}(\Gamma )$, 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.