Stability properties of multiplicative representations of the free group

Iozzi, Alessandra; Kuhn, M.; Steger, Tim

doi:10.1090/tran/7552

Stability properties of multiplicative representations of the free group
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by Alessandra Iozzi, M. Gabriella Kuhn and Tim Steger PDF

Trans. Amer. Math. Soc. 371 (2019), 8699-8731 Request permission

Abstract:

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.

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