CAT(0) Geometry for the Thompson Group

We investigate Farley’s CAT(0) cubical model for Thompson’s group F (we adopt the classical language of F, using binary trees and piecewise linear maps, avoiding the one of diagram groups and pictures). Main results include: in general, Thompson’s group elements are parabolic; we find simple, exact formulas for the CAT(0) translation lengths, in particular the elements of F are ballistic and uniformly bounded away from zero; there exist flats of any dimension and we construct explicitly many of them; we reveal large regions in the Tits Boundary, for example the positive part of a non-separable Hilbert sphere , but also more complicated objects. En route, we solve several open problems proposed in Farley’s papers.