scholarly journals JORGENSEN'S INEQUALITY AND COLLARS IN n-DIMENSIONAL QUATERNIONIC HYPERBOLIC SPACE

2010 ◽  
Vol 62 (3) ◽  
pp. 523-543 ◽  
Author(s):  
W. Cao ◽  
J. R. Parker
Keyword(s):  
Hyperbolic Space ◽  
Quaternionic Hyperbolic Space ◽  
Jørgensen’S Inequality

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.

Naturally reductive homogeneous real hypersurfaces in quaternionic space forms

2003 ◽  
Vol 74 (1) ◽  
pp. 87-100
Author(s):  
Setsuo Nagai
Keyword(s):  
Projective Space ◽  
Hyperbolic Space ◽  
Real Hypersurfaces ◽  
Quaternionic Projective Space ◽  
Space Forms ◽  
The Family ◽  
Quaternionic Hyperbolic Space ◽  
Quaternionic Space ◽  
Naturally Reductive

AbstractWe determine the naturally reductive homogeneous real hypersurfaces in the family of curvature-adapted real hypersurfaces in quaternionic projective space HPn(n ≥ 3). We conclude that the naturally reductive curvature-adapted real hypersurfaces in HPn are Q-quasiumbilical and vice-versa. Further, we study the same problem in quaternionic hyperbolic space HHn(n ≥ 3).

scholarly journals ON DISCRETENESS OF SUBGROUPS OF QUATERNIONIC HYPERBOLIC ISOMETRIES

2019 ◽  
Vol 101 (2) ◽  
pp. 283-293
Author(s):  
KRISHNENDU GONGOPADHYAY ◽  
MUKUND MADHAV MISHRA ◽  
DEVENDRA TIWARI
Keyword(s):  
Linear Group ◽  
Hyperbolic Space ◽  
Dense Subgroup ◽  
Totally Geodesic ◽  
Fixed Element ◽  
Quaternionic Hyperbolic Space

Let $\mathbf{H}_{\mathbb{H}}^{n}$ denote the $n$-dimensional quaternionic hyperbolic space. The linear group $\text{Sp}(n,1)$ acts on $\mathbf{H}_{\mathbb{H}}^{n}$ by isometries. A subgroup $G$ of $\text{Sp}(n,1)$ is called Zariski dense if it neither fixes a point on $\mathbf{H}_{\mathbb{H}}^{n}\cup \unicode[STIX]{x2202}\mathbf{H}_{\mathbb{H}}^{n}$ nor preserves a totally geodesic subspace of $\mathbf{H}_{\mathbb{H}}^{n}$. We prove that a Zariski dense subgroup $G$ of $\text{Sp}(n,1)$ is discrete if for every loxodromic element $g\in G$ the two-generator subgroup $\langle f,gfg^{-1}\rangle$ is discrete, where the generator $f\in \text{Sp}(n,1)$ is a certain fixed element not necessarily from $G$.

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