Combinatorics

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Pure mathematics

Unitary representation

Mathematics

Mathematical physics

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Lie group

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Quantum mechanics

Unitary state

Journal of Group Theory

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>F</m:mi> <m:mo>,</m:mo> <m:mi>F</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:math> {[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.

On the Haagerup and Kazhdan properties of R. Thompson’s groups