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Transactions of the American Mathematical Society

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Stability properties of multiplicative representations of the free group


Authors: Alessandra Iozzi, M. Gabriella Kuhn and Tim Steger
Journal: Trans. Amer. Math. Soc. 371 (2019), 8699-8731
MSC (2010): Primary 22D10, 43A65; Secondary 15A48, 22E45, 22E40
DOI: https://doi.org/10.1090/tran/7552
Published electronically: March 20, 2019
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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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Additional Information

Alessandra Iozzi
Affiliation: Departement Mathematik, ETH Zürich, 8092 Zürich, Switzerland
Email: iozzi@math.ethz.ch

M. Gabriella Kuhn
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Milano “Bicocca”, Via Cozzi 53, Building U5, 20126 Milano, Italia
Email: mariagabriella.kuhn@unimib.it

Tim Steger
Affiliation: Facoltà di Scienze Matematiche Fisiche e Naturali, Università degli Studi di Sassari, Via Piandanna 4, 07100 Sassari, Italia
Email: steger@uniss.it

DOI: https://doi.org/10.1090/tran/7552
Keywords: Free group, irreducible unitary representation, boundary realization
Received by editor(s): March 3, 2016
Received by editor(s) in revised form: January 15, 2018
Published electronically: March 20, 2019
Additional Notes: The first author was partially supported by the Swiss National Science Foundation project 2000021-127016/2 and 200020-144373
The second and third authors were partially supported by PRIN
Article copyright: © Copyright 2019 American Mathematical Society