CAT(0) geometry for the Thompson Group

TL;DR

This paper explores the CAT(0) geometric structure of Thompson's group F, revealing its elements' parabolic nature, explicit translation lengths, existence of flats of all dimensions, and complex boundary regions, thus advancing understanding of its geometric properties.

Contribution

It provides explicit formulas for translation lengths, constructs flats of arbitrary dimension, and solves open problems related to the CAT(0) geometry of Thompson's group F.

Findings

Thompson's group elements are parabolic in the CAT(0) model.

Explicit formulas for CAT(0) translation lengths are derived.

Existence of flats of any dimension and complex boundary regions is demonstrated.

Abstract

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). 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.