Boundaries of right-angled Coxeter groups with manifold nerves

Abstract

All abstract reflection groups act geometrically on non-positively curved geodesic spaces. Their natural space at infinity, consisting of (bifurcating) infinite geodesic rays emanating from a fixed base point, is called a boundary of the group.We will present a condition on right-angled Coxeter groups under which they have topologically homogeneous boundaries. The condition is that they have a nerve which is a connected closed orientable PL manifold.In the event that the group is generated by the reflections of one of Davis’ exotic open contractible n-manifolds (n⩾4), the group will have a boundary which is a homogeneous cohomology manifold. This group boundary can then be used to equivariantly Z-compactify the Davis manifold.If the compactified manifold is doubled along the group boundary, one obtains a sphere if n⩾5. The system of reflections extends naturally to this sphere and can be augmented by a reflection whose fixed point set is the group boundary. It will be shown that the fixed point set of each extended original reflection on the thus formed sphere is a tame codimension-one sphere.