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

Abstract

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 conjugationinvariant 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 (F H) 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.

Infinite-dimensional Polish groups and Property (T)

“…Two conspicuous concrete cases of our theorem are the groups Aut(µ) and Aut * (µ) mentioned above. Pestov [Pes18,§9] lists the question of whether these groups are Kazhdan as open. In the case of Aut(µ), however, it seems likely that Property (T) could be deduced from the work of Neretin [Ner92][Ner96, §8.4] on the representations of this group, in a similar fashion as Bekka's proof for U ( 2 ) from the works of Kirillov and Ol'shanski (although the works [Ner92,Ner96] only contain a description of the irreducible representations of Aut(µ), and an argument has to be added in order to complete the classification of all unitary representations).…”

“…In a recent work, Pestov [Pes18] showed that amenability and (strong) Property (T) remain contradictory within the class of unitarily representable SIN groups, thus providing several non-examples of strong Property (T) in the non-locally compact setting. In particular, if G is a non-trivial, compact, metrizable group, the Polish group L 0 ([0, 1], G) of random elements of G does not admit a finite Kazhdan set.…”

“…Inspired by Solecki's work, Pestov proved moreover that L 0 ([0, 1], T ) does not admit any compact Kazhdan set. The last section of [Pes18] provides a nice summary of the current state of knowledge concerning Kazhdan's (T) and related properties for several important examples of infinite-dimensional topological groups.…”

“…We can actually deduce the following, which in particular confirms, for PRP groups, a suspicion of Bekka [Bek03,p. 512] that was disproved by Pestov [Pes18,§8] for arbitrary topological groups.…”

On the First Order Cohomology of Infinite-Dimensional Unitary Groups

“…In particular, Theorem 7.9 reveals that the group U p (H) (with 1 ≤ p < ∞) neither has property (FH) nor has property (T). For p = 2, the fact that U 2 (H) does not have property (T) has already been shown in [Pe17]. The question whether U 2 (H) has property (FH) is however stated there as an open problem (see [Pe17,3.5]) which is answered by Theorem 7.9.…”