Coarse Alexander Duality and Duality Groups

We study discrete group actions on coarse Poincare duality spaces, e.g. acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups. When G is an (n − 1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplicial action G y X determines a collection of “peripheral” subgroups H1, . . . , Hk ⊂ G so that the group pair (G, {H1, . . . , Hk}) is an n-dimensional Poincare duality pair. In particular, if G is a 2-dimensional 1-ended group of type FP2, and G y 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.