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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.

MSC:
20 Group theory and generalizations
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