Groups Acting on Hyperbolic Space: Harmonic Analysis and Number Theory
Springer Science & Business Media, Nov 12, 1997 - Mathematics - 524 pages
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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assume asymptotic binary hermitian forms bounded Chapter choose cocompact groups cofinite cofinite group computation congruence subgroup conjugate constant contained converges Corollary corresponding cusps of F define Definition denote Dirichlet series discontinuous groups discrete group discrete subgroup dv(Q eigenfunctions eigenpackets eigenvalues Eisenstein series Euclidean finite covolume finite index fixed points Fourier expansion functional equation fundamental domain geodesic group F Hence hermitian forms Hilbert space holomorphic hyperbolic or loxodromic hyperbolic space imaginary quadratic number implies integral invariant Iso(IH isometries isomorphism Kleinian model Laplace operator lattice Lemma Let F linear loxodromic loxodromic element matrix meromorphic continuation non-zero norm notation obtain orthogonal Poincare polyhedron prime ideal proof of Theorem Proposition prove quadratic form quadratic number field quaternion algebra result Riemann right-hand side ring of integers satisfies Section Selberg zeta function self-adjoint T-invariant tetrahedron Theorem 5.1 theory vector zero
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Page 505 - Automorphic forms, Representation theory and Arithmetic. Tata Institute of Fundamental Research, Bombay, 41-115 Harder, G.
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Page 505 - Proc. Int. Colloq. On Discrete Subgroups of Lie Groups and Applications to Moduli at Bombay (1973), Oxford University Press (1975), 129-160.
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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).