## Coarse Alexander duality and duality groups (1999)

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Venue: | JOURNAL OF DIFFERENTIAL GEOMETRY |

Citations: | 18 - 5 self |

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@ARTICLE{Kapovich99coarsealexander,

author = {Michael Kapovich and Bruce Kleiner},

title = {Coarse Alexander duality and duality groups },

journal = {JOURNAL OF DIFFERENTIAL GEOMETRY},

year = {1999},

volume = {69},

pages = {279--352}

}

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### 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 F P2, and G � X is a free simplicial action on a coarse P D(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 P D(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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