On the Haagerup and Kazhdan properties of R. Thompson’s groups

Abstract

A machinery developed by the second author produces a rich family of unitary representations of the Thompson groups F, T and V. We use it to give direct proofs of two previously known results. First, we exhibit a unitary representation of V that has an almost invariant vector but no nonzero {[F,F]} -invariant vectors reproving and extending Reznikoff’s result that any intermediate subgroup between the commutator subgroup of F and V does not have Kazhdan’s property (T) (though Reznikoff proved it for subgroups of T). Second, we construct a one parameter family interpolating between the trivial and the left regular representations of V. We exhibit a net of coefficients for those representations which vanish at infinity on T and converge to 1 thus reproving that T has the Haagerup property after Farley who further proved that V has this property.

Haagerup property for wreath products constructed with Thompson's groups

“…The argument works identically for countable and uncountable discrete groups Γ. Interestingly, the coefficients of Thompson's group V appearing in step one are not the one constructed by Farley nor the one previously constructed by the author and Jones but coincide when we restrict those coefficients to the smaller Thompson's group T , see Remark 3.8, [Far03,BJ19a].…”

“…Using Schoenberg Theorem applied to the square of the norm of this cocycle we obtain a one parameter family of positive definite maps f α : V Ñ C, 0 ă α ă 1 satisfying that f α pvq " α 2npvq´2 where npvq is the minimum number of leaves for which v is described by a fraction of symmetric trees with npvq leaves. In [BJ19a], Jones and the author constructed a family of positive definite maps on V that coincide with the maps of Farley when restricted to Thompson's group T , see [BJ19a, Remark 1], but do not vanishes at infinity on the group V . A similar observation shows that the restriction to T of our maps φ α coincide with the maps of Farley.…”

“…The action remembers some structure of the category D and, in particular, if the target category is the category of Hilbert spaces (with isometries for morphisms), then the action π Φ is a unitary representation. This provides large families of unitary representations of the Thompson's groups [BJ19a,BJ19b,ABC19,Jon19]. We name π Φ : G C ñ X Φ Jones' actions and more specifically Jones' representations if D is the category of Hilbert spaces.…”

“…There have been increasing results on analytical properties of Thompson's groups F ď T ď V : Reznikoff showed that Thompson's group T does not have Kazhdan's Property (T) and Farley proved that the larger Thompson's group V has the Haagerup property [Rez01,Far03]. Using Jones' actions, Jones and the author have built a net of positive definite coefficients on V that converges pointwise to one and vanish at infinity but only for the intermediate Thompson's group T thus reproving partially the result of Farley [BJ19a]. Using again Jones' machinery we define a new net of coefficients that now vanish at infinity on the larger group V .…”

Pythagorean representations of Thompson's groups

“…Using tensor products instead of direct products we built families of representations having coefficients vanishing at infinity in [BJ18]. Proposition 5.1 demonstrates how different those representations are from the Pythagorean one.…”