Groups Acting on Hyperbolic Space: Harmonic Analysis and Number Theory

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Springer Science & Business Media, Nov 12, 1997 - Mathematics - 524 pages
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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 with well-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,
 

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Page 502 - Expansion in eigenfunctions of the Laplace operator on the fundamental domain of a discrete group on the Lobachevskv plane, Trudy Moskov.
Page 507 - G. HUMBERT. Sur les représentations propres d'un entier par les formes positives d'Hermite, dans un corps quadratique imaginaire (p.
Page 505 - Automorphic forms, Representation theory and Arithmetic. Tata Institute of Fundamental Research, Bombay, 41-115 Harder, G.
Page 505 - Hatcher, A. (1983): Hyperbolic structures of arithmetic type on some link complements. J. London Math. Soc. 27, 345-355 Heath-Brown, DR, Patterson, SJ (1979): The distribution of Kummer sums at prime arguments.
Page 505 - Proc. Int. Colloq. On Discrete Subgroups of Lie Groups and Applications to Moduli at Bombay (1973), Oxford University Press (1975), 129-160.
Page 512 - SPECIAL LIMIT POINTS FOR FUCHSIAN GROUPS AND AUTOMORPHIC FUNCTIONS NEAR THE LIMIT SET. INDIANA UNIV. MATH. J. 24, 143-148 I 1974) 285.30014 NICHOLLS RL.
Page 502 - J. Fischer [An Approach to the Selberg Trace Formula via the Selberg Zeta-Function (Lect. Notes Math. 1253) (Springer 1987; Zbl. 61 8.1 0029 ) ] but it is stated to work for a larger class of, not necessarily even, test functions. Remarkably, in the formulation not only a 'determinant' related to the hyperbolic surface X occurs, but also the corresponding quantity for the 2-sphere.
Page 498 - Modular Functions and Dirichlet Series in Number Theory, Graduate Texts in Math.
Page 506 - Über einen Zusammenhang zwischen der Spektraltheorie automorpher Funktionen auf dem oberen Halbraum und den Klassenzahlen biquadratischer Zahlkörper. (On a connection between the spectral theory of automorphic functions on the upper half-space and the class numbers of biquadratic number fields). Diss. (German). Schriftenreihe des Mathematischen Instituts der Universität Münster. 3. Serie. 17. Münster: Univ., Math. Inst. 80 p. (1994).

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About the author (1997)

Elstrodt-University of Munster, Russia

Grunewald-University of Bonn, Germany

Mennicke-University of Bielefeld, Germany

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