Proper Isometric Actions of Thompson’s Groups on Hilbert Space

In 1965, Thompson defined the groups F, T , and V [8]. Thompson’s group V is the group of right-continuous bijections v of [0, 1] that map dyadic rational numbers to dyadic rational numbers, that are differentiable except at finitely many dyadic rational numbers, and such that, on each interval on which v is differentiable, v is affine with derivative a power of 2. The group F is the subgroup of V consisting of homeomorphisms. The group T is the subgroup of V consisting of those elements which induce homeomorphisms of the circle, where the circle is regarded as [0, 1] with 0 and 1 identified. It is a long-standing open question to determine whether F is amenable. The main theorem of this paper establishes that the groups F, T , and V all have the weaker property of a-T-menability. A theorem of Higson and Kasparov [4] states that every a-T-menable group satisfies the Baum-Connes conjecture with arbitrary coefficients, so Thompson’s groups F, T , and V satisfy the Baum-Connes conjecture as well. An isometric action of a discrete group G on a metric space X is proper if, for any x ∈ X and any bounded subset U of X, there are only finitely many elements of g that translate x inside U. A function f : V1 → V2 between two vector spaces is affine if it is the composition of a linear map followed by a translation.