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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
DOI: https://doi.org/10.1090/proc/12685
Published electronically: July 24, 2015
MathSciNet review: 3415574
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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.


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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

DOI: https://doi.org/10.1090/proc/12685
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