Coarse alexander duality and duality groups

Michael Kapovich, Bruce Kleiner

Research output: Contribution to journalArticle

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, . . ., Hk ⊂ G 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 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.

Original languageEnglish (US)
Pages (from-to)279-352
Number of pages74
JournalJournal of Differential Geometry
Volume69
Issue number2
DOIs
StatePublished - Jan 1 2005

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

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