Geometric two-dimensional duality groups.

*(English)*Zbl 1347.57005The 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.

} 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.

Reviewer: Daniele Ettore Otera (Vilnius)

##### 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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