Discontinuous groups of isometries in the hyperbolic plane. Edited by Asmus L. Schmidt
type: Item
instance of: scholarly article
title: Discontinuous groups of isometries in the hyperbolic plane. Edited by Asmus L. Schmidt
zbMATH Open document ID: 1022.51016
published in: De Gruyter Studies in Mathematics
publication date: 2003-02-23T00:00:00Z
review text: 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 di…
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 51-02
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 30-02
Mathematics Subject Classification ID: 57M60
zbMATH DE Number: 1873123
zbMATH Keywords: hyperbolic geometry
zbMATH Keywords: discontinuous group
zbMATH Keywords: discrete group
zbMATH Keywords: isometry
zbMATH Keywords: hyperbolic plane
zbMATH Keywords: non-Euclidean geometry
zbMATH Keywords: surface automorphism
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
author: Werner Fenchel
author: Jakob Nielsen
description: scientific article; zbMATH DE number 1873123