LEADER 04935nam 2200685Ia 450 001 9910809760103321 005 20240516172054.0 010 $a3-11-089135-2 024 7 $a10.1515/9783110891355 035 $a(CKB)2670000000251071 035 $a(EBL)934895 035 $a(SSID)ssj0000595053 035 $a(PQKBManifestationID)11354592 035 $a(PQKBTitleCode)TC0000595053 035 $a(PQKBWorkID)10554699 035 $a(PQKB)10846214 035 $a(MiAaPQ)EBC934895 035 $a(WaSeSS)Ind00010152 035 $a(DE-B1597)56098 035 $a(OCoLC)840444354 035 $a(OCoLC)842286424 035 $a(DE-B1597)9783110891355 035 $a(Au-PeEL)EBL934895 035 $a(CaPaEBR)ebr10597994 035 $a(OCoLC)843204843 035 $a(PPN)175503281 035 $a(EXLCZ)992670000000251071 100 $a20021003d2003 uy 0 101 0 $aeng 135 $aurnn#---|u||u 181 $ctxt 182 $cc 183 $acr 200 10$aDiscontinuous groups of isometries in the hyperbolic plane$b[electronic resource] /$fWerner Fenchel, Jakob Nielsen ; edited by Asmus L. Schmidt 205 $a1st ed. 210 $aBerlin ;$aNew York $cWalter de Gruyter$d2003 215 $a1 online resource (388 p.) 225 0 $aDe Gruyter studies in mathematics ;$v29 300 $aDescription based upon print version of record. 311 0 $a3-11-017526-6 320 $aIncludes bibliographical references (p. [355]-359) and index. 327 $tFront matter --$tChapter I. Möbius transformations and non-euclidean geometry. --$t§1 Pencils of circles - inversive geometry --$t§2 Cross-ratio --$t§3 Möbius transformations, direct and reversed --$t§4 Invariant points and classification of Möbius transformations --$t§5 Complex distance of two pairs of points --$t§6 Non-euclidean metric --$t§7 Isometric transformations --$t§8 Non-euclidean trigonometry --$t§9 Products and commutators of motions --$tChapter II. Discontinuous groups of motions and reversions. --$t§10 The concept of discontinuity --$t§11 Groups with invariant points or lines --$t§12 A discontinuity theorem --$t§13 ?-groups. Fundamental set and limit set --$t§14 The convex domain of an ?-group. Characteristic and isometric neighbourhood --$t§15 Quasi-compactness modulo ? and finite generation of ? --$tChapter III. Surfaces associated with discontinuous groups. --$t§16 The surfaces D modulo ? and K(?) modulo ? --$t§17 Area and type numbers --$tChapter IV. Decompositions of groups. --$t§18 Composition of groups --$t§19 Decomposition of groups --$t§20 Decompositions of ?-groups containing reflections --$t§21 Elementary groups and elementary surfaces --$t§22 Complete decomposition and normal form in the case of quasi-compactness --$t§23 Exhaustion in the case of non-quasi-compactness --$tChapter V. Isomorphism and homeomorphism. --$t§24 Topological and geometrical isomorphism --$t§25 Topological and geometrical homeomorphism --$t§26 Construction of g-mappings. Metric parameters. Congruent groups --$tSymbols and definitions --$tAlphabets --$tBibliography --$tIndex 330 $aThis is an introductory textbook on isometry groups of the hyperbolic plane. Interest in such groups dates back more than 120 years. Examples appear in number theory (modular groups and triangle groups), the theory of elliptic functions, and the theory of linear differential equations in the complex domain (giving rise to the alternative name Fuchsian groups). The current book is based on what became known as the famous Fenchel-Nielsen manuscript. Jakob Nielsen (1890-1959) started this project well before World War II, and his interest arose through his deep investigations on the topology of Riemann surfaces and from the fact that the fundamental group of a surface of genus greater than one is represented by such a discontinuous group. Werner Fenchel (1905-1988) joined the project later and overtook much of the preparation of the manuscript. The present book is special because of its very complete treatment of groups containing reversions and because it avoids the use of matrices to represent Moebius maps. This text is intended for students and researchers in the many areas of mathematics that involve the use of discontinuous groups. 410 3$aDe Gruyter Studies in Mathematics 606 $aDiscontinuous groups 606 $aIsometrics (Mathematics) 615 0$aDiscontinuous groups. 615 0$aIsometrics (Mathematics) 676 $a514/.2 686 $aSK 380$2rvk 700 $aFenchel$b W$g(Werner),$f1905-$0492545 701 $aNielsen$b Jakob$f1890-1959.$0737717 701 $aSchmidt$b Asmus L$01599041 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910809760103321 996 $aDiscontinuous groups of isometries in the hyperbolic plane$93921585 997 $aUNINA