Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Property (T) and the Furstenberg entropy of nonsingular actions

Authors: Lewis Bowen, Yair Hartman and Omer Tamuz
Journal: Proc. Amer. Math. Soc. 144 (2016), 31-39
MSC (2010): Primary 20F69, 37A40
Published electronically: July 24, 2015
MathSciNet review: 3415574
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We establish a new characterization of property (T) in terms of the Furstenberg entropy of nonsingular actions. Given any generating measure $ \mu $ on a countable group $ G$, A. Nevo showed that a necessary condition for $ G$ to have property (T) is that the Furstenberg $ \mu $-entropy values of the ergodic, properly nonsingular $ G$-actions are bounded away from zero. We show that this is also a sufficient condition.

References [Enhancements On Off] (What's this?)

  • [1] B. Bekka, P. de la Harpe, and A. Valette,
    Kazhdan's property (T),
    Cambridge University Press, 2008.
  • [2] L. Bowen, Random walks on random coset spaces with applications to Furstenberg entropy, Invent. Math. 196 (2014), no. 2, 485-510. MR 3193754,
  • [3] L. Bowen, Y. Hartman, and O. Tamuz,
    Generic stationary measures and actions,
    arXiv preprint arXiv:1405.2260, 2014.
  • [4] A. Connes and B. Weiss, Property $ {\rm T}$ and asymptotically invariant sequences, Israel J. Math. 37 (1980), no. 3, 209-210. MR 599455 (82e:28023b),
  • [5] H. Furstenberg, Noncommuting random products, Trans. Amer. Math. Soc. 108 (1963), 377-428. MR 0163345 (29 #648)
  • [6] Y. Hartman and O. Tamuz,
    Furstenberg entropy realizations for virtually free groups and lamplighter groups,
    Journal d'Analyse Mathématique, to appear, 2015.
  • [7] V. F. R. Jones and Klaus Schmidt, Asymptotically invariant sequences and approximate finiteness, Amer. J. Math. 109 (1987), no. 1, 91-114. MR 878200 (88h:28021),
  • [8] V. A. Kaĭmanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Probab. 11 (1983), no. 3, 457-490. MR 704539 (85d:60024)
  • [9] D. A. Každan, On the connection of the dual space of a group with the structure of its closed subgroups, Funkcional. Anal. i Priložen. 1 (1967), 71-74 (Russian). MR 0209390 (35 #288)
  • [10] A. Nevo, The spectral theory of amenable actions and invariants of discrete groups, Geom. Dedicata 100 (2003), 187-218. MR 2011122 (2004j:22012),
  • [11] A. Nevo and R. J. Zimmer, Rigidity of Furstenberg entropy for semisimple Lie group actions, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 3, 321-343 (English, with English and French summaries). MR 1775184 (2001k:22009),
  • [12] N. Ozawa,
    Noncommutative real algebraic geometry of Kazhdan's property (T),
    J. Inst. Math. Jussieu, FirstView:1-6, 11 2014.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 20F69, 37A40

Retrieve articles in all journals with MSC (2010): 20F69, 37A40

Additional Information

Lewis Bowen
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712

Yair Hartman
Affiliation: Department of Mathematics, Weizmann Institute of Science, 761001 Rehovot, Israel

Omer Tamuz
Affiliation: Microsoft Research, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Received by editor(s): June 30, 2014
Received by editor(s) in revised form: December 1, 2014
Published electronically: July 24, 2015
Additional Notes: The first author was supported in part by NSF grant DMS-0968762, NSF CAREER Award DMS-0954606 and BSF grant 2008274.
The second author was supported by the European Research Council, grant 239885
Communicated by: Nimish Shah
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society