04935nam 2200685Ia 450 991080976010332120240516172054.03-11-089135-210.1515/9783110891355(CKB)2670000000251071(EBL)934895(SSID)ssj0000595053(PQKBManifestationID)11354592(PQKBTitleCode)TC0000595053(PQKBWorkID)10554699(PQKB)10846214(MiAaPQ)EBC934895(WaSeSS)Ind00010152(DE-B1597)56098(OCoLC)840444354(OCoLC)842286424(DE-B1597)9783110891355(Au-PeEL)EBL934895(CaPaEBR)ebr10597994(OCoLC)843204843(PPN)175503281(EXLCZ)99267000000025107120021003d2003 uy 0engurnn#---|u||utxtccrDiscontinuous groups of isometries in the hyperbolic plane[electronic resource] /Werner Fenchel, Jakob Nielsen ; edited by Asmus L. Schmidt1st ed.Berlin ;New York Walter de Gruyter20031 online resource (388 p.)De Gruyter studies in mathematics ;29Description based upon print version of record.3-11-017526-6 Includes bibliographical references (p. [355]-359) and index.Front matter --Chapter I. Möbius transformations and non-euclidean geometry. --§1 Pencils of circles - inversive geometry --§2 Cross-ratio --§3 Möbius transformations, direct and reversed --§4 Invariant points and classification of Möbius transformations --§5 Complex distance of two pairs of points --§6 Non-euclidean metric --§7 Isometric transformations --§8 Non-euclidean trigonometry --§9 Products and commutators of motions --Chapter II. Discontinuous groups of motions and reversions. --§10 The concept of discontinuity --§11 Groups with invariant points or lines --§12 A discontinuity theorem --§13 ℱ-groups. Fundamental set and limit set --§14 The convex domain of an ℱ-group. Characteristic and isometric neighbourhood --§15 Quasi-compactness modulo ℱ and finite generation of ℱ --Chapter III. Surfaces associated with discontinuous groups. --§16 The surfaces D modulo ℭ and K(ℱ) modulo ℱ --§17 Area and type numbers --Chapter IV. Decompositions of groups. --§18 Composition of groups --§19 Decomposition of groups --§20 Decompositions of ℱ-groups containing reflections --§21 Elementary groups and elementary surfaces --§22 Complete decomposition and normal form in the case of quasi-compactness --§23 Exhaustion in the case of non-quasi-compactness --Chapter V. Isomorphism and homeomorphism. --§24 Topological and geometrical isomorphism --§25 Topological and geometrical homeomorphism --§26 Construction of g-mappings. Metric parameters. Congruent groups --Symbols and definitions --Alphabets --Bibliography --IndexThis is an introductory textbook on isometry groups of the hyperbolic plane. Interest in such groups dates back more than 120 years. Examples appear in number theory (modular groups and triangle groups), the theory of elliptic functions, and the theory of linear differential equations in the complex domain (giving rise to the alternative name Fuchsian groups). The current book is based on what became known as the famous Fenchel-Nielsen manuscript. Jakob Nielsen (1890-1959) started this project well before World War II, and his interest arose through his deep investigations on the topology of Riemann surfaces and from the fact that the fundamental group of a surface of genus greater than one is represented by such a discontinuous group. Werner Fenchel (1905-1988) joined the project later and overtook much of the preparation of the manuscript. The present book is special because of its very complete treatment of groups containing reversions and because it avoids the use of matrices to represent Moebius maps. This text is intended for students and researchers in the many areas of mathematics that involve the use of discontinuous groups.De Gruyter Studies in MathematicsDiscontinuous groupsIsometrics (Mathematics)Discontinuous groups.Isometrics (Mathematics)514/.2SK 380rvkFenchel W(Werner),1905-492545Nielsen Jakob1890-1959.737717Schmidt Asmus L1599041MiAaPQMiAaPQMiAaPQBOOK9910809760103321Discontinuous groups of isometries in the hyperbolic plane3921585UNINA