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On the Haagerup and Kazhdan properties of R. Thompson’s groups. (English) Zbl 07104291
Summary: 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.

20 Group theory and generalizations
Full Text: DOI arXiv
[1] C. A. Akemann and M. E. Walter, Unbounded negative definite functions, Canad. J. Math. 33 (1981), no. 4, 862-871. · Zbl 0437.22004
[2] J. W. Cannon, W. J. Floyd and W. R. Parry, Introductory notes on Richard Thompson’s groups, Enseign. Math. (2) 42 (1996), no. 3-4, 215-256. · Zbl 0880.20027
[3] D. S. Farley, Finiteness and \rm CAT(0) properties of diagram groups, Topology 42 (2003), no. 5, 1065-1082. · Zbl 1044.20023
[4] D. S. Farley, Proper isometric actions of Thompson’s groups on Hilbert space, Int. Math. Res. Not. (2003), no. 45, 2409-2414. · Zbl 1113.22005
[5] D. S. Farley, Actions of picture groups on CAT(0) cubical complexes, Geom. Dedicata 110 (2005), 221-242. · Zbl 1139.20038
[6] E. Ghys and V. Sergiescu, Sur un groupe remarquable de difféomorphismes du cercle, Comment. Math. Helv. 62 (1987), no. 2, 185-239. · Zbl 0647.58009
[7] V. F. R. Jones, A no-go theorem for the continuum limit of a periodic quantum spin chain, Comm. Math. Phys. 357 (2018), no. 1, 295-317. · Zbl 1397.82025
[8] D. A. Každan, On the connection of the dual space of a group with the structure of its closed subgroups, Funktsional. Anal. i Prilozhen. 1 (1967), 71-74.
[9] A. I. Mal’cev, Nilpotent semigroups, Ivanov. Gos. Ped. Inst. Uč. Zap. Fiz.-Mat. Nauki 4 (1953), 107-111.
[10] A. Navas, Actions de groupes de Kazhdan sur le cercle, Ann. Sci. Éc. Norm. Supér. (4) 35 (2002), no. 5, 749-758. · Zbl 1028.58010
[11] A. Reznikov, Analytic topology, European Congress of Mathematics. Vol. I (Barcelona 2000), Progr. Math. 201, Birkhäuser, Basel (2001), 519-532. · Zbl 1035.53042
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