Includes bibliographical references (p. [355]-359) and index.

Editor's preface - Short biography of the authors - Mobius transformations and non-euclidean geometry - Pencils of circles - Inversive geometry - Cross-ratio - Mobius tranformations, direct and reversed - Invariant points and classification of Mobius transformations - Complex distance of two pairs of points - Non-Euclidian metric - Geometric transformations - Non-Euclidean trigonometry - Products and commutators of motions - Discontinuous groups of motions and reversions - The concept of discontinuity - Groups with invariant points or lines - A discontinuity theorem - F-groups. Fundamental set and limit set - The Convex domain of F-group. Characteristic and isometric neighbourhood - Quasi-compactness modulo F and finite generation of F - Surfaces associated with discontinuous groups - The surfaces D modulo G and K(F) modulo F - Area and type numbers - Decompositions of groups - Composition of groups - Decomposition of groups - Decompositions of F-groups containing reflections - Elementary groups and elementary surfaces - Complete decomposition and normal form in the case of quasi-compactness - Exhaustion in the case of non-quasi-compactness - Isomorphism and homeomorphism - Topological and geometrical isomorphism - Topological and geometrical homeomorphism - Construction of g-mappings. Metric parameters. Congruent groups - Symbols and definitions - Bibliography

Fuchsian groups play a central role in various important fields of mathematics. This text is based on what became known as the Fenchel-Nielsen manuscript. It covers such topics as mobius transformations and surfaces associated with discontinous groups.