Polyhedral Preliminaries

This chapter deals with cell complexes (also called “polyhedra”) with metrics of piecewise constant curvature. In other words, each cell is required to be isometric to a convex polytope in a space of constant curvature \(\kappa \) . As \(\kappa =-1, 0, +1\) such metrics are called, respectively, piecewise hyperbolic, piecewise euclidean, or piecewise spherical. A geodesic metric space is \(\operatorname {CAT}(\kappa )\) if geodesic triangles in it satisfy Gromov’s comparison inequality of Cartan, Aleksandrov and Toponogov with respect to triangles in a plane of constant curvature \(\kappa \) . The fundamental result is that a piecewise constant curvature polyhedron is locally \(\operatorname {CAT}(\kappa )\) if and only if the Link Condition holds for each of its cells, i.e., if each such link is \(\operatorname {CAT}(1)\) . The link of a cell in a cube complex is a piecewise spherical complex with “all right” simplices. Gromov’s Lemma asserts that such a link is \(\operatorname {CAT}(1)\) if and only if it is a flag complex. In Sect. 2.3 we state two standard results about group actions on \(\operatorname {CAT}(0)\) spaces, the Bruhat–Tits Fixed Point Theorem and the Flat Torus Theorem. Some examples of nonpositively curved polygons of groups are explained in Sect. 2.4: Higman groups in Sect. 2.4.3, Burger–Mozes groups in Sect. 2.4.4 and nonpositively curved polygons of groups in Sect. 2.4.5.