Stability properties of multiplicative representations of the free group
HTML articles powered by AMS MathViewer
- 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.
References
- Hyman Bass and Ravi Kulkarni, Uniform tree lattices, J. Amer. Math. Soc. 3 (1990), no. 4, 843–902. MR 1065928, DOI 10.1090/S0894-0347-1990-1065928-2
- Hyman Bass and Alexander Lubotzky, Tree lattices, Progress in Mathematics, vol. 176, Birkhäuser Boston, Inc., Boston, MA, 2001. With appendices by Bass, L. Carbone, Lubotzky, G. Rosenberg and J. Tits. MR 1794898, DOI 10.1007/978-1-4612-2098-5
- M. Cowling and T. Steger, The irreducibility of restrictions of unitary representations to lattices, J. Reine Angew. Math. 420 (1991), 85–98. MR 1124567
- Alessandro Figà-Talamanca and Massimo A. Picardello, Spherical functions and harmonic analysis on free groups, J. Functional Analysis 47 (1982), no. 3, 281–304. MR 665019, DOI 10.1016/0022-1236(82)90108-2
- Alessandro Figà-Talamanca and Massimo A. Picardello, Restriction of spherical representations of $\textrm {PGL}_{2}(\textbf {Q}_{p})$ to a discrete subgroup, Proc. Amer. Math. Soc. 91 (1984), no. 3, 405–408. MR 744639, DOI 10.1090/S0002-9939-1984-0744639-3
- Alessandra Iozzi, M. Gabriella Kuhn, and Tim Steger, A new family of representations of virtually free groups, Math. Z. 274 (2013), no. 1-2, 167–184. MR 3054322, DOI 10.1007/s00209-012-1062-4
- M. Gabriella Kuhn and Tim Steger, Monotony of certain free group representations, J. Funct. Anal. 179 (2001), no. 1, 1–17. MR 1807250, DOI 10.1006/jfan.2000.3663
- M. Gabriella Kuhn and Tim Steger, Free group representations from vector-valued multiplicative functions. I, Israel J. Math. 144 (2004), 317–341. MR 2121544, DOI 10.1007/BF02916716
- Frank Thomson Leighton, Finite common coverings of graphs, J. Combin. Theory Ser. B 33 (1982), no. 3, 231–238. MR 693362, DOI 10.1016/0095-8956(82)90042-9
- Alexander Lubotzky, Lattices in rank one Lie groups over local fields, Geom. Funct. Anal. 1 (1991), no. 4, 406–431. MR 1132296, DOI 10.1007/BF01895641
- George W. Mackey, The theory of unitary group representations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1976. Based on notes by James M. G. Fell and David B. Lowdenslager of lectures given at the University of Chicago, Chicago, Ill., 1955. MR 0396826
- William S. Massey, Algebraic topology: an introduction, Graduate Texts in Mathematics, Vol. 56, Springer-Verlag, New York-Heidelberg, 1977. Reprint of the 1967 edition. MR 0448331
- Robert T. Powers, Simplicity of the $C^{\ast }$-algebra associated with the free group on two generators, Duke Math. J. 42 (1975), 151–156. MR 374334
- Carlo Pensavalle and Tim Steger, Tensor products with anisotropic principal series representations of free groups, Pacific J. Math. 173 (1996), no. 1, 181–202. MR 1387798
Additional Information
- Alessandra Iozzi
- Affiliation: Departement Mathematik, ETH Zürich, 8092 Zürich, Switzerland
- MR Author ID: 199039
- 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
- MR Author ID: 248120
- Email: steger@uniss.it
- 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 - © Copyright 2019 American Mathematical Society
- 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
- MathSciNet review: 3955561