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

TL;DR

This paper explores the relationship between amenability and property (T) in non-locally compact topological groups, providing new examples and distinctions, especially within Polish and SIN groups, and examining their representations.

Contribution

It introduces new examples of topological groups with property (T) that challenge previous assumptions and clarifies the interplay between amenability, property (T), and representations in various classes of groups.

Findings

Amenability and property (T) are not mutually exclusive in non-locally compact groups.

Constructed a minimally almost periodic group with property (T) and no finite Kazhdan set.

Showed that property (T) is preserved under arbitrary infinite products.

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.