Discontinuous Groups of Isometries in the Hyperbolic Plane / / Werner Fenchel, Jakob Nielsen; ed. by Asmus L. Schmidt.
This 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...
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Fenchel, Werner, author. aut http://id.loc.gov/vocabulary/relators/aut Discontinuous Groups of Isometries in the Hyperbolic Plane / Werner Fenchel, Jakob Nielsen; ed. by Asmus L. Schmidt. Berlin ; Boston : De Gruyter, [2011] ©2002 1 online resource (364 p.) text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda De Gruyter Studies in Mathematics , 0179-0986 ; 29 Frontmatter -- 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 -- Index restricted access http://purl.org/coar/access_right/c_16ec online access with authorization star This 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. Issued also in print. Mode of access: Internet via World Wide Web. In English. Description based on online resource; title from PDF title page (publisher's Web site, viewed 28. Feb 2023) Discontinuous groups. Isometrics (Mathematics). Hyperbolische Geometrie. Isometriegruppe. Riemannsche Fläche. MATHEMATICS / General. bisacsh Nielsen, Jakob, author. aut http://id.loc.gov/vocabulary/relators/aut Schmidt, Asmus L., editor. edt http://id.loc.gov/vocabulary/relators/edt Title is part of eBook package: De Gruyter DG Studies in Mathematics eBook-Package 9783110494938 ZDB-23-GSM Title is part of eBook package: De Gruyter DGBA Backlist Complete English Language 2000-2014 PART1 9783110238570 Title is part of eBook package: De Gruyter DGBA Backlist Mathematics 2000-2014 (EN) 9783110238471 Title is part of eBook package: De Gruyter DGBA Mathematics - 2000 - 2014 9783110637205 ZDB-23-GMA Title is part of eBook package: De Gruyter E-DITION: BEST OF MATHEMATICS 9783110233957 ZDB-23-DGQ print 9783110175264 https://doi.org/10.1515/9783110891355 https://www.degruyter.com/isbn/9783110891355 Cover https://www.degruyter.com/document/cover/isbn/9783110891355/original |
language |
English |
format |
eBook |
author |
Fenchel, Werner, Fenchel, Werner, Nielsen, Jakob, |
spellingShingle |
Fenchel, Werner, Fenchel, Werner, Nielsen, Jakob, Discontinuous Groups of Isometries in the Hyperbolic Plane / De Gruyter Studies in Mathematics , Frontmatter -- 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 -- Index |
author_facet |
Fenchel, Werner, Fenchel, Werner, Nielsen, Jakob, Nielsen, Jakob, Nielsen, Jakob, Schmidt, Asmus L., Schmidt, Asmus L., |
author_variant |
w f wf w f wf j n jn |
author_role |
VerfasserIn VerfasserIn VerfasserIn |
author2 |
Nielsen, Jakob, Nielsen, Jakob, Schmidt, Asmus L., Schmidt, Asmus L., |
author2_variant |
j n jn a l s al als a l s al als |
author2_role |
VerfasserIn VerfasserIn HerausgeberIn HerausgeberIn |
author_sort |
Fenchel, Werner, |
title |
Discontinuous Groups of Isometries in the Hyperbolic Plane / |
title_full |
Discontinuous Groups of Isometries in the Hyperbolic Plane / Werner Fenchel, Jakob Nielsen; ed. by Asmus L. Schmidt. |
title_fullStr |
Discontinuous Groups of Isometries in the Hyperbolic Plane / Werner Fenchel, Jakob Nielsen; ed. by Asmus L. Schmidt. |
title_full_unstemmed |
Discontinuous Groups of Isometries in the Hyperbolic Plane / Werner Fenchel, Jakob Nielsen; ed. by Asmus L. Schmidt. |
title_auth |
Discontinuous Groups of Isometries in the Hyperbolic Plane / |
title_alt |
Frontmatter -- 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 -- Index |
title_new |
Discontinuous Groups of Isometries in the Hyperbolic Plane / |
title_sort |
discontinuous groups of isometries in the hyperbolic plane / |
series |
De Gruyter Studies in Mathematics , |
series2 |
De Gruyter Studies in Mathematics , |
publisher |
De Gruyter, |
publishDate |
2011 |
physical |
1 online resource (364 p.) Issued also in print. |
contents |
Frontmatter -- 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 -- Index |
isbn |
9783110891355 9783110494938 9783110238570 9783110238471 9783110637205 9783110233957 9783110175264 |
issn |
0179-0986 ; |
callnumber-first |
Q - Science |
callnumber-subject |
QA - Mathematics |
callnumber-label |
QA612 |
callnumber-sort |
QA 3612.14 F46 42003EB |
url |
https://doi.org/10.1515/9783110891355 https://www.degruyter.com/isbn/9783110891355 https://www.degruyter.com/document/cover/isbn/9783110891355/original |
illustrated |
Not Illustrated |
doi_str_mv |
10.1515/9783110891355 |
oclc_num |
840444354 |
work_keys_str_mv |
AT fenchelwerner discontinuousgroupsofisometriesinthehyperbolicplane AT nielsenjakob discontinuousgroupsofisometriesinthehyperbolicplane AT schmidtasmusl discontinuousgroupsofisometriesinthehyperbolicplane |
status_str |
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ids_txt_mv |
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Discontinuous Groups of Isometries in the Hyperbolic Plane / |
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