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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Amenability versus property $(T)$ for non-locally compact topological groups
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by Vladimir G. Pestov PDF
Trans. Amer. Math. Soc. 370 (2018), 7417-7436 Request permission

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
  • Email: vpest283@uottawa.ca
  • 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.
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 7417-7436
  • MSC (2010): Primary 22A25, 43A65, 57S99
  • DOI: https://doi.org/10.1090/tran/7256
  • MathSciNet review: 3841853