Coarse Alexander duality for pairs and applications

For a group G (of type F ) acting properly on a coarse Poincar duality space X, Kapovich and Kleiner introduced a coarse version of Alexander duality between G and its complement in X.More precisely, the cohomology of G with group ring coefficients is dual to a certain ech homology group of the family of increasing neighborhoods of a G-orbit in X.This duality applies more generally to coarse embeddings of certain contractible simplicial complexes into coarse PD.n/ spaces.In this paper we introduce a relative version of this ech homology that satisfies the Eilenberg-Steenrod exactness axiom, and we prove a relative version of coarse Alexander duality.As an application we provide a detailed proof of the following result, first stated by Kapovich and Kleiner.Given a 2-complex formed by gluing k halfplanes along their boundary lines and a coarse embedding into a contractible 3-manifold, the complement consists of k deep components that are arranged cyclically in a pattern called a Jordan cycle.We use the Jordan cycle as an invariant in proving the existence of a 3-manifold group that is virtually Kleinian but not itself Kleinian.