Geometric two-dimensional duality groups (Q2016094)

The present paper deals with connected, finite, aspherical 2-dimensional simplicial complexes which are \textit{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. {\vskip 2mm} 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 \textit{duality group}, which means that the universal cover \(\widetilde \Delta\) satisfies Poincaré duality. {\vskip 2mm} 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.