Groups Acting on Hyperbolic Space

1. Front Matter

   Pages I-XV

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2. Three-Dimensional Hyperbolic Space

   • Jürgen Elstrodt, Fritz Grunewald, Jens Mennicke

   Pages 1-32

3. Groups Acting Discontinuously on Three-Dimensional Hyperbolic Space

   • Jürgen Elstrodt, Fritz Grunewald, Jens Mennicke

   Pages 33-81

4. Automorphic Functions

   • Jürgen Elstrodt, Fritz Grunewald, Jens Mennicke

   Pages 83-129

5. Spectral Theory of the Laplace Operator

   • Jürgen Elstrodt, Fritz Grunewald, Jens Mennicke

   Pages 131-183

6. Spectral Theory of the Laplace Operator for Cocompact Groups

   • Jürgen Elstrodt, Fritz Grunewald, Jens Mennicke

   Pages 185-229

7. Spectral Theory of the Laplace Operator for Cofinite Groups

   • Jürgen Elstrodt, Fritz Grunewald, Jens Mennicke

   Pages 231-310

8. PSL(2) over Rings of Imaginary Quadratic Integers

   • Jürgen Elstrodt, Fritz Grunewald, Jens Mennicke

   Pages 311-357

9. Eisenstein Series for PSL(2) over Imaginary Quadratic Integers

   • Jürgen Elstrodt, Fritz Grunewald, Jens Mennicke

   Pages 359-405

10. Integral Binary Hermitian Forms

    • Jürgen Elstrodt, Fritz Grunewald, Jens Mennicke

    Pages 407-441

11. Examples of Discontinuous Groups

    • Jürgen Elstrodt, Fritz Grunewald, Jens Mennicke

    Pages 443-495

12. Back Matter

    Pages 497-524

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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,

• calculus
• discontinuous group
• eisenstein series
• harmonic analysis
• hermitian form
• hyperbolic space
• number theory
• trace formula

• Mathematisches Institut, Universität Münster, Münster, Germany

  Jürgen Elstrodt

• Mathematisches Institut, Universität Düsseldorf, Düsseldorf, Germany

  Fritz Grunewald

• Fakultät für Mathematik, Universität Bielefeld, Bielefeld, Germany

  Jens Mennicke

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