On the Haagerup and Kazhdan properties of R. Thompson’s groups. (English) Zbl 1515.20212

Brothier, Arnaud; Jones, Vaughan F. R.

J. Group Theory 22, No. 5, 795-807 (2019).

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.

MSC:

20F65	Geometric group theory
46L80	\(K\)-theory and operator algebras (including cyclic theory)

References:

[1]	C. A. Akemann and M. E. Walter, Unbounded negative definite functions, Canad. J. Math. 33 (1981), no. 4, 862-871.; Akemann, C. A.; Walter, M. E., Unbounded negative definite functions, Canad. J. Math., 33, 4, 862-871 (1981) · 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.; Cannon, J. W.; Floyd, W. J.; Parry, W. R., Introductory notes on Richard Thompson’s groups, Enseign. Math. (2), 42, 3-4, 215-256 (1996) · Zbl 0880.20027
[3]	D. S. Farley, Finiteness and \rm CAT(0) properties of diagram groups, Topology 42 (2003), no. 5, 1065-1082.; Farley, D. S., Finiteness and \rm CAT(0) properties of diagram groups, Topology, 42, 5, 1065-1082 (2003) · 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.; Farley, D. S., Proper isometric actions of Thompson’s groups on Hilbert space, Int. Math. Res. Not., 45, 2409-2414 (2003) · Zbl 1113.22005
[5]	D. S. Farley, Actions of picture groups on CAT(0) cubical complexes, Geom. Dedicata 110 (2005), 221-242.; Farley, D. S., Actions of picture groups on CAT(0) cubical complexes, Geom. Dedicata, 110, 221-242 (2005) · 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.; Ghys, E.; Sergiescu, V., Sur un groupe remarquable de difféomorphismes du cercle, Comment. Math. Helv., 62, 2, 185-239 (1987) · 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.; Jones, V. F. R., A no-go theorem for the continuum limit of a periodic quantum spin chain, Comm. Math. Phys., 357, 1, 295-317 (2018) · 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.; Každan, D. A., On the connection of the dual space of a group with the structure of its closed subgroups, Funktsional. Anal. i Prilozhen., 1, 71-74 (1967) · Zbl 0168.27602
[9]	A. I. Mal’cev, Nilpotent semigroups, Ivanov. Gos. Ped. Inst. Uč. Zap. Fiz.-Mat. Nauki 4 (1953), 107-111.; Mal’cev, A. I., Nilpotent semigroups, Ivanov. Gos. Ped. Inst. Uč. Zap. Fiz.-Mat. Nauki, 4, 107-111 (1953)
[10]	A. Navas, Actions de groupes de Kazhdan sur le cercle, Ann. Sci. Éc. Norm. Supér. (4) 35 (2002), no. 5, 749-758.; Navas, A., Actions de groupes de Kazhdan sur le cercle, Ann. Sci. Éc. Norm. Supér. (4), 35, 5, 749-758 (2002) · Zbl 1028.58010
[11]	A. Reznikov, Analytic topology, European Congress of Mathematics. Vol. I (Barcelona 2000), Progr. Math. 201, Birkhäuser, Basel (2001), 519-532.; Reznikov, A., Analytic topology, European Congress of Mathematics. Vol. I, 519-532 (2001) · Zbl 1035.53042