Eigenfunctions of the laplacian on a real hyperbolic space

AbstractLet ℍ be the division ring of real quaternions. Let SL(2, ℍ) be the group of 2 × 2 quaternionic matrices $A={\scriptsize{(\begin{array}{l@{\quad}l} a & b \\ c & d \end{array})}}$ with quaternionic determinant det A = |ad − aca−1b| = 1. This group acts by the orientation-preserving isometries of the five-dimensional real hyperbolic space. We obtain discreteness criteria for Zariski-dense subgroups of SL(2, ℍ).

Download Full-text

Isometric immersions in the hyperbolic space with their image contained in a horoball

Glasgow Mathematical Journal ◽

10.1017/s0017089501010011 ◽

2001 ◽

Vol 43 (1) ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Ana Lluch

Keyword(s):

Hyperbolic Space ◽

Mean Curvature ◽

Curvature Vector ◽

Complete Riemannian Manifold ◽

Isometric Immersions ◽

Special Groups ◽

The Mean ◽

Complex Hyperbolic Spaces ◽

Real Hyperbolic Space ◽

Sharp Lower Bound

We give a sharp lower bound for the supremum of the norm of the mean curvature of an isometric immersion of a complete Riemannian manifold with scalar curvature bounded from below into a horoball of a complex or real hyperbolic space. We also characterize the horospheres of the real or complex hyperbolic spaces as the only isometrically immersed hypersurfaces which are between two parallel horospheres, have the norm of the mean curvature vector bounded by the above sharp bound and have some special groups of symmetries.

Download Full-text

Tensor Square of the Minimal Representation of O(p, q)

Canadian Mathematical Bulletin ◽

10.4153/cmb-2007-005-x ◽

2007 ◽

Vol 50 (1) ◽

pp. 48-55 ◽

Cited By ~ 2

Author(s):

Alexander Dvorsky

Keyword(s):

Tensor Product ◽

Hyperbolic Space ◽

Irreducible Representations ◽

Plancherel Measure ◽

Minimal Representation ◽

Direct Integral ◽

Tensor Square ◽

Real Hyperbolic Space

AbstractIn this paper, we study the tensor product π = σmin ⊗ σmin of the minimal representation σmin of O(p, q) with itself, and decompose π into a direct integral of irreducible representations. The decomposition is given in terms of the Plancherel measure on a certain real hyperbolic space.

Download Full-text

Jørgensen’s Inequality and Algebraic Convergence Theorem in Quaternionic Hyperbolic Isometry Groups

Abstract and Applied Analysis ◽

10.1155/2014/684594 ◽

2014 ◽

Vol 2014 ◽

pp. 1-5

Author(s):

Huani Qin ◽

Yueping Jiang ◽

Wensheng Cao

Keyword(s):

Hyperbolic Space ◽

Convergence Theorem ◽

Kleinian Group ◽

Isometry Groups ◽

Quaternionic Hyperbolic Space ◽

Algebraic Convergence ◽

Real Hyperbolic Space ◽

Jørgensen’S Inequality ◽

Algebraic Limit

We obtain an analogue of Jørgensen's inequality in quaternionic hyperbolic space. As an application, we prove that if ther-generator quaternionic Kleinian group satisfies I-condition, then its algebraic limit is also a quaternionic Kleinian group. Our results are generalizations of the counterparts in then-dimensional real hyperbolic space.

Download Full-text

Kähler groups, real hyperbolic spaces and the Cremona group. With an appendix by Serge Cantat

Compositio Mathematica ◽

10.1112/s0010437x11007068 ◽

2011 ◽

Vol 148 (1) ◽

pp. 153-184 ◽

Cited By ~ 7

Author(s):

Thomas Delzant ◽

Pierre Py

Keyword(s):

Hyperbolic Space ◽

Discrete Subgroup ◽

Hyperbolic Spaces ◽

Classical Theorem ◽

Isometric Action ◽

Cremona Group ◽

The Real ◽

Infinite Dimensional ◽

Kähler Groups ◽

Real Hyperbolic Space

AbstractGeneralizing a classical theorem of Carlson and Toledo, we prove thatanyZariski dense isometric action of a Kähler group on the real hyperbolic space of dimension at least three factors through a homomorphism onto a cocompact discrete subgroup of PSL2(ℝ). We also study actions of Kähler groups on infinite-dimensional real hyperbolic spaces, describe some exotic actions of PSL2(ℝ) on these spaces, and give an application to the study of the Cremona group.

Download Full-text