Hyperbolic n-Manifolds

Abstract

In this chapter, we study hyperbolic n-manifolds. We begin with a geometric method for constructing spherical, Euclidean, and hyperbolic n-manifolds. In Section 11.2, we prove Poincaré’s fundamental polyhedron theorem for freely acting groups. In Section 11.3, we prove the Gauss-Bonnet theorem. In Section 11.4, we determine the simplices of maximum volume in hyperbolic n-space. In Section 11.5, we study differential forms. In Section 11.6, we introduce the Gromov norm of a closed hyperbolic manifold. In Section 11.7, we study measure homology. In Section 11.8, we prove Mostow’s rigidity theorem for closed hyperbolic manifolds.