Celebratio Mathematica — Jones

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Works connected to Arnaud Brothier

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A. Brothier and V. F. R. Jones: “Hilbert modules over a planar algebra and the Haagerup property,” J. Funct. Anal. 269 : 11 (December 2015), pp. 3634–3644. MR 3406863 Zbl 1339.46058 ArXiv 1503.02708 article

A. Brothier and V. F. R. Jones: “Pythagorean representations of Thompson’s groups,” J. Funct. Anal. 277 : 7 (October 2019), pp. 2442–2469. MR 3989149 Zbl 07089431 ArXiv 1807.06215 article

A. Brothier and V. F. R. Jones: “On the Haagerup and Kazhdan properties of R. Thompson’s groups,” J. Group Theory 22 : 5 (2019), pp. 795–807. MR 4000616 Zbl 07104291 ArXiv 1805.02177 article

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.