scholarly journals Discreteness Criteria of Möbius Groups of High Dimensions and Convergence Theorems of Kleinian Groups

2001 ◽  
Vol 159 (1) ◽  
pp. 68-82 ◽  
Author(s):  
Xiantao Wang ◽  
Weiqi Yang
Keyword(s):  
Kleinian Groups ◽  
High Dimensions ◽  
Convergence Theorems ◽  
Discreteness Criteria ◽  
Möbius Groups

KLEINIAN GROUPS WITH REAL PARAMETERS

2001 ◽  
Vol 03 (02) ◽  
pp. 163-186 ◽  
Author(s):  
F. W. GEHRING ◽  
J. P. GILMAN ◽  
G. J. MARTIN
Keyword(s):  
General Structure ◽  
Kleinian Groups ◽  
Analytic Space ◽  
Fuchsian Groups ◽  
Discrete Groups ◽  
Möbius Groups

We find all real points of the analytic space of two generator Möbius groups with one generator elliptic of order two. Geometrically this is a certain slice through the space of two generator discrete groups, analogous to the Riley slice, though of a very different nature. We obtain applications concerning the general structure of the space of all two generator Kleinian groups and various universal constraints for Fuchsian groups.

scholarly journals Discreteness and Convergence of Complex Hyperbolic Isometry Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Xi Fu
Keyword(s):  
Isometry Groups ◽  
Convergence Theorems ◽  
Discreteness Criteria ◽  
Algebraic Convergence

We investigate the discreteness and convergence of complex isometry groups and some discreteness criteria and algebraic convergence theorems for subgroups ofPU(n,1)are obtained. All of the results are generalizations of the corresponding known ones.

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 ALGEBRAIC CONVERGENCE THEOREMS OF COMPLEX KLEINIAN GROUPS

2012 ◽  
Vol 55 (1) ◽  
pp. 1-8 ◽  
Author(s):  
WENSHENG CAO
Keyword(s):  
Kleinian Groups ◽  
Limit Group ◽  
Convergence Theorems ◽  
Möbius Group ◽  
Algebraic Convergence ◽  
Algebraic Limit

AbstractLet {Gr,i} be a sequence of r-generator subgroups of U(1,n; ℂ) and Gr be its algebraic limit group. In this paper, two algebraic convergence theorems concerning {Gr,i} and Gr are obtained. Our results are generalisations of their counterparts in the n-dimensional sense-preserving Möbius group.

scholarly journals A two-dimensional slice through the parameter space of two-generator Kleinian groups

2018 ◽  
Vol 28 (08) ◽  
pp. 1535-1564
Author(s):  
Elena Klimenko ◽  
Natalia Kopteva
Keyword(s):  
Parameter Space ◽  
Kleinian Groups ◽  
Two Dimensional ◽  
Real Trace ◽  
Discreteness Criteria

We describe all real points of the parameter space of two-generator Kleinian groups with a parabolic generator, that is, we describe a certain two-dimensional slice through this space. In order to do this, we gather together known discreteness criteria for two-generator groups with a parabolic generator and present them in the form of conditions on parameters. We complete the description by giving discreteness criteria for groups generated by a parabolic and a [Formula: see text]-loxodromic elements whose commutator has real trace and present all orbifolds uniformized by such 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.

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