Coxeter groups act on CAT(0) cube complexes

Abstract

We show that any finitely generated Coxeter group acts properly discontinuously on a locally finite, finite-dimensional CAT(0) cube complex. For any word-hyperbolic or right-angled Coxeter group we prove that the cubing is cocompact. We show how the local structure of the cubing is related to the partial order studied by Brink and Howlett in their proof of automaticity for Coxeter groups.