The Geometry of Discrete Groups

Description

This text is about the geometric theory of discrete groups and the associated tesselations of the underlying space. The theory of Möbius transformations in n-dimensional Euclidean space is developed. These transformations are discussed as isometries of hyperbolic space and are then identified with the elementary transformations of complex analysis. A detailed account of analytic hyperbolic trigonometry is given, and this forms the basis of the subsequent analysis of tesselations of the hyperbolic plane. Emphasis is placed on the geometrical aspects of the subject and on the universal constraints which must be satisfied by all tesselations.;1 Preliminary Material -- 2 Matrices -- 3 Möbius Transformations on?n -- 4 Complex Möbius Transformations -- 5 Discontinuous Groups -- 6 Riemann Surfaces -- 7 Hyperbolic Geometry -- 8 Fuchsian Groups -- 9 Fundamental Domains -- 10 Finitely Generated Groups -- 11 Universal Constraints on Fuchsian Groups -- References. 1 Preliminary Material -- 2 Matrices -- 3 Möbius Transformations on?n -- 4 Complex Möbius Transformations -- 5 Discontinuous Groups -- 6 Riemann Surfaces -- 7 Hyperbolic Geometry -- 8 Fuchsian Groups -- 9 Fundamental Domains -- 10 Finitely Generated Groups -- 11 Universal Constraints on Fuchsian Groups -- References.