Groups Acting on Hyperbolic Space

Harmonic Analysis and Number Theory
1. Auflage. Softcover version of original hardcover Auflage 1998
ISBN: 978-3-642-08302-0
Verlag: Springer

Buch, Englisch, 524 Seiten, Previously published in hardcover, Format (B × H): 155 mm x 235 mm, Gewicht: 814 g

Reihe: Springer Monographs in Mathematics

This book is concerned with discontinuous groups of motions of the unique connected and simply connected Riemannian 3-manifold of constant curva ture -1, which is traditionally called hyperbolic 3-space. This space is the 3-dimensional instance of an analogous Riemannian manifold which exists uniquely in every dimension n: 2. The hyperbolic spaces appeared first in the work of Lobachevski in the first half of the 19th century. Very early in the last century the group of isometries of these spaces was studied by Steiner, when he looked at the group generated by the inversions in spheres. The ge ometries underlying the hyperbolic spaces were of fundamental importance since Lobachevski, Bolyai and Gauß had observed that they do not satisfy the axiom of parallels. Already in the classical works several concrete coordinate models of hy perbolic 3-space have appeared. They make explicit computations possible and also give identifications of the full group of motions or isometries withwell-known matrix groups. One such model, due to H. Poincare, is the upper 3 half-space IH in JR. The group of isometries is then identified with an exten sion of index 2 of the group PSL(2,

Zielgruppe

Research

Autoren/Hrsg.

Elstrodt, Juergen

Mennicke, Jens

Grunewald, Fritz

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Zahlentheorie
• Mathematik | Informatik Mathematik Algebra Algebraische Strukturen, Gruppentheorie
• Mathematik | Informatik Mathematik Mathematische Analysis Harmonische Analysis, Fourier-Mathematik

Inhaltsverzeichnis

1. Three-Dimensional Hyperbolic Space.- 2. Groups Acting Discontinuously on Three-Dimensional Hyperbolic Space.- 3. Automorphic Functions.- 4. Spectral Theory of the Laplace Operator.- 5. Spectral Theory of the Laplace Operator for Cocompact Groups.- 6. Spectral Theory of the Laplace Operator for Cofinite Groups.- 7. PSL(2) over Rings of Imaginary Quadratic Integers.- 8. Eisenstein Series for PSL(2) over Imaginary Quadratic Integers.- 9. Integral Binary Hermitian Forms.- 10. Examples of Discontinuous Groups.- References.