Coarse Alexander duality and duality groups

Abstract

We study discrete group actions on coarse Poincaré duality spaces, e.g., acyclic simplicial complexes which admit free cocompact group actions by Poincaré duality groups. When G is an (n−1) dimensional duality group and X is a coarse Poincaré duality space of formal dimension n, then a free simplicial action G ↷ X determines a collection of “peripheral” subgroups H₁, … H_k ⊂ G so that the group pair (G, {H₁,…H_k }) is an n-dimensional Poincaré duality pair. In particular, if G is a 2-dimensional 1-ended group of type FP₂, and G ↷ X is a free simplicial action on a coarse PD(3) space X, then G contains surface subgroups; if in addition X is simply connected, then we obtain a partial generalization of the Scott/Shalen compact core theorem to the setting of coarse PD(3) spaces. In the process, we develop coarse topological language and a formulation of coarse Alexander duality which is suitable for applications involving quasi-isometries and geometric group theory.