TEST MAP AND DISCRETENESS IN SL(2, ℍ)

2018 ◽  
Vol 61 (03) ◽  
pp. 523-533 ◽  
Author(s):  
KRISHNENDU GONGOPADHYAY ◽  
ABHISHEK MUKHERJEE ◽  
SUJIT KUMAR SARDAR
Keyword(s):  
Hyperbolic Space ◽  
Division Ring ◽  
Discreteness Criteria ◽  
Dense Subgroups ◽  
Real Quaternions ◽  
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).

scholarly journals DISCRETENESS CRITERIA FOR ISOMETRIC GROUPS ACTING ON COMPLEX HYPERBOLIC SPACES

2010 ◽  
Vol 81 (3) ◽  
pp. 481-487
Author(s):  
XI FU
Keyword(s):  
Hyperbolic Space ◽  
Complex Hyperbolic Space ◽  
Hyperbolic Spaces ◽  
Discreteness Criteria ◽  
Condition C ◽  
Dense Subgroups ◽  
Complex Hyperbolic Spaces ◽  
Möbius Groups

AbstractIn this paper, four new discreteness criteria for isometric groups on complex hyperbolic spaces are proved, one of which shows that the Condition C hypothesis in Cao [‘Discrete and dense subgroups acting on complex hyperbolic space’, Bull. Aust. Math. Soc.78 (2008), 211–224, Theorem 1.4] is removable; another shows that the parabolic condition hypothesis in Li and Wang [‘Discreteness criteria for Möbius groups acting on $\overline {\mathbb {R}}^n$ II’, Bull. Aust. Math. Soc.80 (2009), 275–290, Theorem 3.1] is not necessary.

scholarly journals DISCRETE AND DENSE SUBGROUPS ACTING ON COMPLEX HYPERBOLIC SPACE

2008 ◽  
Vol 78 (2) ◽  
pp. 211-224 ◽  
Author(s):  
WENSHENG CAO
Keyword(s):  
Hyperbolic Space ◽  
Complex Hyperbolic Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Dense Subgroup ◽  
Discreteness Criteria ◽  
Dense Subgroups ◽  
Necessary And Sufficient

AbstractIn this paper, we study the discreteness criteria for nonelementary subgroups of U(1,n;ℂ) acting on complex hyperbolic space. Several discreteness criteria are obtained. As applications, we obtain a classification of nonelementary subgroups of U(1,n;ℂ) and show that any dense subgroup of SU(1,n;ℂ) contains a dense subgroup generated by at most n elements when n≥2. We also obtain a necessary and sufficient condition for the normalizer of a discrete and nonelementary subgroup in SU(1,n;ℂ) to be discrete.

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

2001 ◽  
Vol 43 (1) ◽  
pp. 1-8 ◽  
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.

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

2007 ◽  
Vol 50 (1) ◽  
pp. 48-55 ◽  
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.

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

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.

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

2011 ◽  
Vol 148 (1) ◽  
pp. 153-184 ◽  
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.

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