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 JØRGENSEN’S INEQUALITY FOR QUATERNIONIC HYPERBOLIC SPACE WITH ELLIPTIC ELEMENTS

2009 ◽  
Vol 81 (1) ◽  
pp. 121-131 ◽  
Author(s):  
WENSHENG CAO ◽  
HAIOU TAN
Keyword(s):  
Hyperbolic Space ◽  
Two Dimensional ◽  
Möbius Group ◽  
Quaternionic Hyperbolic Space ◽  
Groups Of Isometries ◽  
Jørgensen’S Inequality

AbstractIn this paper, we give an analogue of Jørgensen’s inequality for nonelementary groups of isometries of quaternionic hyperbolic space generated by two elements, one of which is elliptic. As an application, we obtain an analogue of Jørgensen’s inequality in the two-dimensional Möbius group of the above case.

scholarly journals ON THE ALGEBRAIC CONVERGENCE OF FINITELY GENERATED KLEINIAN GROUPS IN ALL DIMENSIONS

2011 ◽  
Vol 85 (2) ◽  
pp. 275-279 ◽  
Author(s):  
XI FU
Keyword(s):  
Kleinian Group ◽  
Kleinian Groups ◽  
Finitely Generated ◽  
Limit Group ◽  
Algebraic Convergence ◽  
Algebraic Limit

AbstractLet {Gr,i} be a sequence of r-generator Kleinian groups acting on ${\overline {\mathbb {R}}}^n$. In this paper, we prove that if {Gr,i} satisfies the F-condition, then its algebraic limit group Gr is also a Kleinian group. The existence of a homomorphism from Gr to Gr,i is also proved. These are generalisations of all known corresponding results.

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 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.

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