Nonpositive curvature and reflection groups (Q2776336)

This paper is a survey on polyhedral spaces of nonpositive curvature up to around 1996. For more recent results the reader may consult [\textit{M. W. Davis} and \textit{B. Okun}, Vanishing theorems and conjectures for the \(L^2\)-homology of right-angled Coxeter groups, Geom. Topology 5, No. 2, 7-74 (2001)]. NEWLINENEWLINENEWLINEAfter recalling basic facts of metric geometry, the author concentrates on polyhedra of piecewise constant curvature, for which there exist convenient combinatorial criteria, due to Gromov and Moussong, ensuring that the curvature is bounded above. NEWLINENEWLINENEWLINEThen ideal boundaries of nonpositively curved spaces are discussed along with the notion of an ``infinitesimal shadow'' which measures the non-uniqueness of geodesic continuation at a point. The next section explains how various results from geometric topology help to understand the ``topology at infinity'' of contractible manifolds. NEWLINENEWLINENEWLINEThe next topic is the combinatorial Gauß-Bonnet theorem and the Hopf conjecture on the sign of the Euler characteristic of closed piecewise Euclidean manifolds. This is related to a conjecture on flag triangulations of spheres, and to Singer's conjecture on the vanishing of reduced \({\l}^2\)-cohomology away from the middle dimension. NEWLINENEWLINENEWLINEThe second part of the paper is a detailed discussion of Coxeter and Artin groups in relation with the Hopf conjecture, and Eilenberg-Ganea's problem on the difference between geometric and cohomological dimensions.NEWLINENEWLINEFor the entire collection see [Zbl 0977.00029].