Abstract
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 ↷ 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 ↷ 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.
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