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Geometric two-dimensional duality groups. (English) Zbl 1347.57005
The present paper deals with connected, finite, aspherical 2-dimensional simplicial complexes which are Cohen-Macaulay, namely complexes $$\Delta$$ such that the link of any vertex is non-empty and connected, and the link of any edge is non-empty, too. {
} Since the universal cover of $$\Delta$$ is assumed to be contractible, the simplicial complex $$\Delta$$ is a finite $$K(G,1)$$-complex, where $$G$$ is the group $$\pi_1 (\Delta, v)$$, for $$v$$ a vertex of $$\Delta$$. The main problem is to understand when $$G$$ has one end. In such a case the 2-dimensional group $$G$$ becomes a duality group, which means that the universal cover $$\widetilde \Delta$$ satisfies Poincaré duality. {
} The main result (Theorem 5.6) concerns one-endedness of very general contractible, locally-finite, infinite, 2-dimensional Cohen-Macaulay complexes. As a consequence of this (quite technical) theorem, the author provides necessary and sufficient conditions for $$\widetilde \Delta$$ to be one-ended (Corollary 6.13), and simple combinatorial conditions on $$\Delta$$ assuring that $$G=\pi_1 (\Delta, v)$$ is a 2-dimensional duality group.
##### MSC:
 57M20 Two-dimensional complexes (manifolds) (MSC2010) 20F65 Geometric group theory 55U10 Simplicial sets and complexes in algebraic topology 55U30 Duality in applied homological algebra and category theory (aspects of algebraic topology)
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