Discontinuous groups of isometries in the hyperbolic plane. Edited by Asmus L. Schmidt
This book is a foundational essay on the hyperbolic geometry of the plane. It is written by two eminent geometers, Jacob Nielsen (1890-1959) and Werner Fenchel (1905-1988). Nielsen is the founder of several theories. We can mention here the theory of reduction for systems of generators in a free group, which he developed in the 1920's, and that of the classification of homotopy classes of surface automorphisms, which he developed in the 1930's, and which has been revived several decades later by W. P. Thurson in his work on Teichmüller space and surface mapping class groups. Werner Fenchel made important contributions to the geometric theory of convexity. The book under review developed through lectures given by the two authors for more than 40 years. The book has existed for a long time in the form of mimeographed notes, known as the ``Fenchel-Nielsen manuscript''. A first draft was already planned to appear in the Princeton Mathematical Series in 1948. But then the subject grew up, the authors made several revisions; After the death of Nieslen, Fenchel continued this writing project alone until his death. The book in its actual form is edited by Asmus L. Schmidt, who was a young collaborator of Fenchel. The contents of the book is as follows : Chapter 1 contains basic material on the hyperbolic geometry of the disk: Möbius transformations, cross-ratio, trigonometric formulas, and a discussion on commutators. Chapter 2 concerns discontinuous groups of motions of the disk: fundamental domains, invariant points, lines and axes, criteria for a group to be discontinuous, limit sets and convexity. Chapter 3 concerns surfaces associated with discontinuous groups: type, area, genus, normal domains. Chapter 4 concerns abstract group theory: surface groups as generalized free products, amalgamation, extensions of groups, decomposition of groups, classification of subgroups, and a theory of normal forms. Chapter 5 concerns isomorphism problems between group actions on the disk and their extension to a correspondence between convex hulls. Special isomorphisms are those which preserve concurrent axes. There are characterization of isomorphisms in terms of metric parameters (displacement lengths). The various chapters are not independent, and the book can be read as a long paper.
scientific article; zbMATH DE number 1873123