Non-inner amenability of the Thompson groups T and V

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.

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The homology of the Higman–Thompson groups

Inventiones mathematicae ◽

10.1007/s00222-018-00848-z ◽

2019 ◽

Vol 216 (2) ◽

pp. 445-518 ◽

Cited By ~ 1

Author(s):

Markus Szymik ◽

Nathalie Wahl

Keyword(s):

Thompson Groups

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On Jones’ subgroup of R. Thompson’s group T

International Journal of Algebra and Computation ◽

10.1142/s0218196718500388 ◽

2018 ◽

Vol 28 (05) ◽

pp. 877-903

Author(s):

Jordan Nikkel ◽

Yunxiang Ren

Keyword(s):

Unitary Representation ◽

Unitary Representations ◽

Thompson Groups ◽

Diagram Groups ◽

Thompson’S Group ◽

Planar Algebra ◽

Thompson Group

Jones introduced unitary representations for the Thompson groups [Formula: see text] and [Formula: see text] from a given subfactor planar algebra. Some interesting subgroups arise as the stabilizer of certain vector, in particular the Jones subgroups [Formula: see text] and [Formula: see text]. Golan and Sapir studied [Formula: see text] and identified it as a copy of the Thompson group [Formula: see text]. In this paper, we completely describe [Formula: see text] and show that [Formula: see text] coincides with its commensurator in [Formula: see text], implying that the corresponding unitary representation is irreducible. We also generalize the notion of the Stallings 2-core for diagram groups to [Formula: see text], showing that [Formula: see text] and [Formula: see text] are not isomorphic, but as annular diagram groups they have very similar presentations.

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