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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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##### References:
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