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

Pestov, Vladimir

doi:10.1090/tran/7256

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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