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

TL;DR

This paper uses a machine to produce unitary representations of Thompson groups to give new proofs of known properties, including the absence of Kazhdan's property (T) in certain subgroups and the Haagerup property of T.

Contribution

It introduces a machine-generated family of unitary representations of Thompson groups to provide direct proofs of their Haagerup and Kazhdan properties, offering new insights into their structure.

Findings

Constructed a representation of V with almost invariant vectors but no nonzero [F,F]-invariant vectors.

Reproved that intermediate subgroups between [F,F] and V lack Kazhdan's property (T).

Established that T has the Haagerup property through a converging net of coefficients.

Abstract

A machine 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, at least for T, Reznikov's result that any intermediate subgroup between the commutator subgroup of F and V does not have Kazhdan's property (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 Farley's result that T has the Haagerup property.