Table of Contents

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

About the Author

Asmus L. Schmidt is Associate Professor at the Institute for Mathematical Sciences of the University of Copenhagen, Denmark.

Reviews

"The Fenchel-Nielsen manuscript has been famous for a long time already and its final publication is a valuable edition to mathematical literature."EMS Newsletter "Those working in the field will be grateful to the editor Asmus Schmidt for producing this classic text; it can now be cited without the annoying reference 'Fenchel an Nielsen (to appear)'."David Singerman in: Bulletin of the London Mathematical Society 36/2004