Combinatorial conditions that imply word-hyperbolicity for 3-manifolds (Q1398213)

A triangulation of a closed \(3\)-manifold is called a \textit{\(5/6^*\)-triangulation} if each edge has degree \(5\) or \(6\) and each \(2\)-cell of the triangulation contains at most one edge of degree \(5\). In the paper under review, it is shown that every \(5/6^*\)-triangulation of a closed \(3\)-manifold admits a piecewise Euclidean metric of non positive curvature, and the universal cover contains no isometrically embedded flat planes. The proof consists in a combinatorial analysis of the links of the vertices of a \(5/6^*\)-triangulation, and a check of the link condition (assuring that the complex is non-positively curved). The check is carried out by a computer using a program, developed by the first two authors, which implements an improved version of an algorithm described by the first two authors in [Exp. Math. 11, 143-158 (2002; Zbl 1042.20030)] (the improvement uses a result of \textit{B. H. Bowditch} [Geometric group theory, (Columbus, 1992), de Gruyter, Berlin, 1-48 (1995; Zbl 0865.53035)]). As a consequence of the main result, a conjecture formulated by Thurston is established: the fundamental group of a closed \(3\)-manifold admitting a \(5/6^*\)-triangulation is word-hyperbolic.