Journal of Differential Geometry

Coarse Alexander duality and duality groups

Michael Kapovich and Bruce Kleiner

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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 GX determines a collection of “peripheral” subgroups H1, … HkG so that the group pair (G, {H1,…Hk }) is an n-dimensional Poincaré duality pair. In particular, if G is a 2-dimensional 1-ended group of type FP2, and GX 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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J. Differential Geom., Volume 69, Number 2 (2005), 279-352.

First available in Project Euclid: 15 July 2005

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Kapovich, Michael; Kleiner, Bruce. Coarse Alexander duality and duality groups. J. Differential Geom. 69 (2005), no. 2, 279--352. doi:10.4310/jdg/1121449108.

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