Discontinuous Groups of Isometries in the Hyperbolic Plane /

Discontinuous Groups of Isometries in the Hyperbolic Plane / / Werner Fenchel, Jakob Nielsen; ed. by Asmus L. Schmidt.

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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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Bibliographic Details
Superior document:Title is part of eBook package: De Gruyter DG Studies in Mathematics eBook-Package
VerfasserIn:
Fenchel, Werner,
Nielsen, Jakob,
HerausgeberIn:
Schmidt, Asmus L.,
Place / Publishing House:Berlin ;, Boston : : De Gruyter, , [2011]
©2002
Year of Publication:2011
Language:English
Series:De Gruyter Studies in Mathematics , 29
Online Access:
Physical Description:1 online resource (364 p.)
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Table of 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

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