A parabolic action on a proper, CAT(0) cube complex

TL;DR

This paper provides a counterexample to a conjecture by Farley, showing that certain diagram groups, including Thompson's group F, can act with parabolic isometries on associated infinite-dimensional CAT(0) cube complexes.

Contribution

It demonstrates that Farley's conjecture does not hold in general by constructing explicit examples where the action includes parabolic isometries.

Findings

Counterexample to Farley's conjecture

Thompson's group F acts with parabolic isometries

Infinite-dimensional cube complexes can have non-semisimple isometries

Abstract

We consider diagram groups as defined by V. Guba and M. Sapir. A diagram group G acts on the associated cube complex K by isometries. It is known that if a cube complex L is of a finite dimension then every isometry g of L is semi-simple, i.e. its translation length is realized. It was conjectured by D. S. Farley that in the case of a diagram group G the action of G on the associated cube complex K is by semisimple isometries even when K has an infinite dimension. In this paper we give a counterexample to Farley Conjecture and we show that R. Thompson's group F, considered as a diagram group, has some elements which act as parabolic (not semi-simple) isometries on the associated cube complex.