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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Amenability versus property $(T)$ for non-locally compact topological groups

Author: Vladimir G. Pestov
Journal: Trans. Amer. Math. Soc. 370 (2018), 7417-7436
MSC (2010): Primary 22A25, 43A65, 57S99
Published electronically: July 5, 2018
MathSciNet review: 3841853
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Abstract: For locally compact groups amenability and Kazhdan’s property $(T)$ are mutually exclusive in the sense that a group having both properties is compact. This is no longer true for more general Polish groups. However, a weaker result still holds for SIN groups (topological groups admitting a basis of conjugation-invariant neighbourhoods of identity): if such a group admits sufficiently many unitary representations, then it is precompact as soon as it is amenable and has the strong property $(T)$ (i.e., admits a finite Kazhdan set). If an amenable topological group with property $(T)$ admits a faithful uniformly continuous representation, then it is maximally almost periodic. In particular, an extremely amenable SIN group never has strong property $(T)$, and an extremely amenable subgroup of unitary operators in the uniform topology is never a Kazhdan group. This leads to first examples distinguishing between property $(T)$ and property $(FH)$ in the class of Polish groups. Disproving a 2003 conjecture by Bekka, we construct a complete, separable, minimally almost periodic topological group with property $(T)$ having no finite Kazhdan set. Finally, as a curiosity, we observe that the class of topological groups with property $(T)$ is closed under arbitrary infinite products with the usual product topology.

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Additional Information

Vladimir G. Pestov
Affiliation: Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, Ontario, Canada K1N 6N5 – and – Departamento de Matemática, Universidade Federal de Santa Catarina, Trindade, Florianópolis, SC, 88.040-900, Brazil
MR Author ID: 138420

Received by editor(s): January 31, 2017
Received by editor(s) in revised form: April 4, 2017
Published electronically: July 5, 2018
Additional Notes: The author was Special Visiting Researcher of the program Science Without Borders of CAPES (Brazil), processo 085/2012.
Article copyright: © Copyright 2018 American Mathematical Society