Coxeter Groups Act on Cat(0) Cube Complexes

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. In his thesis Moussong [Mou87] showed that each finitely generated Coxeter group acts properly discontinuously and cocompactly on a CAT(0) simplicial complex. He used this to characterise word hyperbolic Coxeter groups in terms of their natural presentation. Moussong’s construction together with a result of Alonso and Bridson [AB95, Theorem 5.1], implies the solvability of the conjugacy problem for Coxeter groups. Brink and Howlett subsequently used algebraic techniques to show that Coxeter groups are automatic (see [ECH92] for the definition). It remains an open question whether or not Coxeter groups, or, more generally, groups acting properly discontinuously and cocompactly on CAT(0) spaces satisfy the stronger condition of biautomaticity. On the other hand, groups which act properly discontinuously and cocompactly on CAT(0) cube complexes are known to be biautomatic [NR98], and in attempting to use this fact to show that Coxeter groups are biautomatic we were led to the following construction: Theorem 1. If (W, R) is a finite rank Coxeter system then the Coxeter group W acts properly discontinuously by isometries on a locally finite, finite dimensional CAT(0) cube complex, and there is a quasi-isometric embedding of W in X. Although the action is not, in general, cocompact a result of Williams [Wil99] characterises this phenomenon as follows: the action will be cocompact unless the Coxeter group contains infinitely many conjugacy classes of subgroups isomorphic to a triangle group 〈s1, s2, s3 | s1 = s 2 2 = s 2 3 = (s1s2) p = (s2s3) q = (s3s1) r = 1〉, where p, q, r are finite exponents occurring in the standard presentation of the Coxeter group. So we obtain as a corollary: Corollary 1. If (W, R) is a finite rank Coxeter system such that the Coxeter group contains only finitely many conjugacy classes of subgroups isomorphic to triangle groups. Then the group W is bi-automatic. The local finiteness of the cube complex is related to the finiteness of the set of minimal roots as investigated by Brink and Howlett [BH93], which paper solved the “Parallel Walls Conjecture”. In attempting to understand the local structure of our complex we were led to the following conjecture, which, although we have been unable to prove it, seems worth stating: Date: SUBMITTED VERSION 6th September 2002.