REFLECTION GROUPS AND CAT(0) COMPLEXES WITH EXOTIC LOCAL STRUCTURES

Abstract

We show that, in contrast to the situation for the standard complex on which a right angled Coxeter group W acts, there are cocompact W -actions on CAT(0) complexes such that the local topology of the complex is distinctly different from the end topology of W . If X is a contractible n-manifold or homology n-manifold, then, since it satisfies Poincare duality, its cohomology with compact supports, H∗ c (X), is concentrated in dimension n and is isomorphic to Z in that dimension. It follows that the homology at infinity of X is concentrated in dimension n− 1 and is isomorphic to Z in that dimension (see [?] for definitions and references for homology at infinity). More particularly, recall that a simplicial complex is a homology n-manifold if and only if the link of each vertex is a “generalized homology (n−1)-sphere” (i.e., a homology (n− 1)-manifold with the same homology as Sn−1). So, the above argument shows that if the link of each vertex in an aspherical simplicial complex X is a generalized homology (n− 1)-sphere, then the homology at infinity of its universal cover X is concentrated in dimension n − 1 and hence, X is (n − 2)-acyclic at infinity. This shows that hypotheses concerning the local topology of an aspherical space can have implications for the end topology of its universal cover. There are other local-to-asymptotic results for nonpositively curved complexes. If L is a simplicial complex, we let S(L) denote the set of all closed simplices of L, including the empty simplex. Let X be a (locally finite) CAT(0) cubical complex, and for each vertex x ∈ X, let Lx denote its link. If for each vertex x and for each closed simplex σ ∈ S(Lx), Lx−σ is m-connected (resp., m-acyclic), then X is m-connected (resp., m-acyclic) at infinity (see [?] and the references cited there). We call the complexes Lx − σ the punctured links of X. In recent work [?], we have shown there is a close connection between local topology and end topology of the standard complexes on which Coxeter groups act. The nerve of a Coxeter system (W,S) is the simplicial complex L with one vertex for each element of the generating set S and one simplex for each subset of S which generates a finite subgroup of W . As explained in [?], [?] or [?], associated to (W,S) there is natural cell complex, here denoted |W |, such that |W | is a model for EW and such that the link of each of its vertices is isomorphic to L (see [?] for the definition of EG). This implies, for example, that if L is a triangulation of an (n− 1)-sphere, then |W | is a contractible n-manifold. Date: January 15, 2004.