Abstract
Let π be an irreducible unitary representation of a finitely generated non-abelian free group Γ; suppose π is weakly contained in the regular representation. In 2001 the first and third authors conjectured that such a representation must be either odd or monotonous or duplicitous. In 2004 they introduced the class of multiplicative representations: this is a large class of representations obtained by looking at the action of Γ on its Cayley graph. In the second paper of this series we showed that some of the multiplicative representations were monotonous. Here we show that all the other multiplicative representations are either odd or duplicitous. The conjecture is therefore established for multiplicative representations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Boyer and Ł. Garncarek, Asymptotic Schur orthogonality in hyperbolic groups with application to monotony, Transactions of the American Mathematical Society 371 (2019), 6815–6841.
A. Boyer, G. Link and Ch. Pittet, Ergodic boundary representations, Ergodic Theory and Dynamical Systems 39 (2019), 2017–2047.
A. Boyer and L. A. Pinochet, An ergodic theorem for the quasi-regular representation of the free group, Bulletin of the Belgian Mathematical Society. Simon Stevin 24 (2017), 243–255.
C. Cecchini and A. Figà-Talamanca, Projections of uniqueness for Lp (G), Pacific Journal of Mathematics 51 (1974), 37–47.
K. R. Davidson, C⋆-algebras by Example, Fields Institute Monographs, Vol. 6, American Mathematical Society, Providence, RI, 1996.
A. Figà-Talamanca and A. M. Picardello, Spherical functions and harmonic analysis on free groups, Journal of Functional Analysis 47 (1982), 281–304.
A. Figà-Talamanca and T. Steger, Harmonic analysis for anisotropic random walks on homogeneous trees, Memoirs of the American Mathematical Society 531 (1994).
U. Haagerup, An example of a nonnuclear C⋆-algebra which has the metric approximation property, Inventiones Mathematicae 50 (1979), 279–293.
W. Hebisch, M. G. Kuhn and T. Steger, Free group representations: Duplicity and oddity on the boundary, Transactions of the American Mathematical Society 375 (2022), 1825–1860.
A. Iozzi, M. G. Kuhn and T. Steger, Stability properties of multiplicative representations of the free group, Transactions of the American Mathematical Society 371 (2019), 8699–8731.
M. G. Kuhn, S. Saliani and T. Steger, Free group representations from vector-valued multiplicative functions, II, Mathematische Zeitschrift 284 (2016), 1137–1162.
M. G. Kuhn and T. Steger, Multiplicative functions on free groups and irreducible representations, Pacific Journal of Mathematics 169 (1995), 311–334.
M. G. Kuhn and T. Steger, More irreducible boundary representations of free groups, Duke Mathematical Journal 82 (1996), 381–436.
M. G. Kuhn and T. Steger, Monotony of certain free group representations, Journal of Functional Analysis 179 (2001), 1–17.
M. G. Kuhn and T. Steger, Free group representations from vector-valued multiplicative functions, I, Israel Journal of Mathematics 144 (2004), 317–341.
L. De Michele and A. Figà-Talamanca, Positive definite functions on free groups, American Journal of Mathematics 102 (1980), 503–509.
W. L. Paschke, Pure eigenstates for the sum of generators of the free group, Pacific Journal of Mathematics 197 (2001), 151–171.
W. L. Paschke, Some irreducible free group representations in which a linear combination of the generators has an eigenvalue, Journal of the Australian Mathematical Society 72 (2002), 257–286.
C. Pensavalle and T. Steger, Tensor products with anisotropic principal series representations of free groups, Pacific Journal of Mathematics 173 (1996), 181–202.
T. Pytlik and R. Szwarc, An analytic family of uniformly bounded representations of free groups, Acta Mathematica 157 (1986), 287–309.
W. Rudin, Functional Analysis, International Series in Pure and Applied Mathematics, McGraw-Hill, New York, 1991.
M. Takesaki, Theory of Operator Algebras II, Encyclopaedia of Mathematical Sciences, Vol. 125, Springer, Berlin, 2003.
H. Yoshizawa, Some remarks on unitary representations of the free group, Osaka Mathematical Journal 3 (1951), 55–63.
Acknowledgements
We would like to thank the anonymous reviewer whose constructive comments/suggestions helped improve and clarify the manuscript.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kuhn, M.G., Saliani, S. & Steger, T. Free group representations from vector-valued multiplicative functions. III. Isr. J. Math. 258, 339–373 (2023). https://doi.org/10.1007/s11856-023-2474-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-023-2474-z