Isometries of Hyperbolic Space

Abstract

In this chapter, we study the topology of the group I(Hⁿ) of isometries of hyperbolic space. The chapter begins with an introduction to topological groups. The topological group structure of I(Hⁿ) is studied from various points of view in Section 5.2. The discrete subgroups of I(Hⁿ) are of fundamental importance for the study of hyperbolic manifolds. The basic properties of the discrete subgroups of I(Hⁿ) are examined in Section 5.3. A characterization of the discrete subgroups of I(Eⁿ) is given in Section 5.4. The chapter ends with a characterization of all the elementary discrete subgroups of I(Hⁿ).