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

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.

TEST MAP AND DISCRETENESS IN SL(2, ℍ)

Let 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).

Discreteness and Convergence of Complex Hyperbolic Isometry Groups

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 ISOMETRIC GROUPS ACTING ON COMPLEX HYPERBOLIC SPACES

In 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.

DISCRETENESS CRITERIA FOR MÖBIUS GROUPS ACTING ON II

Jørgensen’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.

DISCRETE AND DENSE SUBGROUPS ACTING ON COMPLEX HYPERBOLIC SPACE

In 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.