Discreteness criteria for Möbius groups acting on $$\mathcal{O}\mathcal{L}_p $$

2005 ◽  
Vol 150 (1) ◽  
pp. 357-368 ◽  
Author(s):  
Xiantao Wang ◽  
Liulan Li ◽  
Wensheng Cao
Keyword(s):  
Discreteness Criteria ◽  
Möbius Groups

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 DISCRETENESS CRITERIA FOR MÖBIUS GROUPS ACTING ON II

2009 ◽  
Vol 80 (2) ◽  
pp. 275-290 ◽  
Author(s):  
LIU-LAN LI ◽  
XIAN-TAO WANG
Keyword(s):  
Fixed Point ◽  
Necessary Conditions ◽  
Careful Analysis ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Point Sets ◽  
Discreteness Criteria ◽  
Möbius Groups ◽  
Jørgensen’S Inequality ◽  
Fixed Point Sets

AbstractJørgensen’s famous inequality gives a necessary condition for a subgroup of PSL(2,ℂ) to be discrete. It is also true that if Jørgensen’s inequality holds for every nonelementary two-generator subgroup, the group is discrete. The sufficient condition has been generalized to many settings. In this paper, we continue the work of Wang, Li and Cao (‘Discreteness criteria for Möbius groups acting on $\overline {\mathbb {R}}^n$’, Israel J. Math.150 (2005), 357–368) and find three more (infinite) discreteness criteria for groups acting on $\overline {\mathbb {R}}^n$; we also correct a linguistic ambiguity of their Theorem 3.3 where one of the necessary conditions might be vacuously fulfilled. The results of this paper are obtained by using known results regarding two-generator subgroups and a careful analysis of the relation among the fixed point sets of various elements of the group.

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).

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