Jørgensen's Inequality for Complex Hyperbolic Space

Jørgensen's inequality gives a necessary condition for a nonelementary two generator group of isometries of hyperbolic space to be discrete. We give analogues of Jørgensen's inequality for nonelementary groups of isometries of complex hyperbolic 2-space generated by two elements, one of which is either loxodromic or boundary elliptic. These results give an improvement over earlier results of Basmajian and Miner.     

Discreteness of subgroups of PU(2,1) with regular elliptic elements

In [Y.-P. Jiang, S. Kamiya, J.R. Parker, Jøgensen’s inequality for complex hyperbolic space, Geometriae Dedicate 97 (2003) 55-80], Jiang et al. provided Jørgensen’s inequality for non-elementary group of isometries of complex hyperbolic space generated by two elements, one of which is loxodromic or boundary elliptic. In this paper, we give analogues of Jørgensen’s inequality for the subgroup generated by two elements, but one of which is regular elliptic.     

Algebraic characterization of the isometries of the hyperbolic 5-space

Let \mathbb GL(2, \mathbbH){{\mathbb G}L(2, \mathbb{H})} be the group of invertible 2 × 2 matrices over the division algebra \mathbbH{\mathbb{H}} of quaternions. \mathbb GL(2, \mathbbH){{\mathbb G}L(2, \mathbb{H})} acts on the hyperbolic 5-space as the group of orientation-preserving isometries. Using this action we give an algebraic characterization of the orientation-preserving isometries of the hyperbolic 5-space. Along the way we also determine the conjugacy classes and the conjugacy classes of centralizers or the z-classes in \mathbb GL(2, \mathbbH){{\mathbb G}L(2, \mathbb{H})} .     

Packings by translation balls in {\widetilde{{\rm SL}_2({\mathbb{R}})}}

For one of Thurston model spaces, \({\widetilde{{\rm SL}_2({\mathbb{R}})}}\) , we discuss translation balls and packing that space by such balls in contrast to the packing by standard (geodesic) balls. We present an infinite family of packings generated by discrete groups of isometries, and observe numerical results on their densities. In particular, we found packings whose densities are close to the upper bound density for ball packings in the hyperbolic 3-space.     

Holonomy groups of five dimensional Bieberbach groups

Let Γ be a Bieberbach group—that is a torsion free crystallographic group. In this paper we give a list of the isomorphism types of all holonomy groups of five-dimensional Bieberbach groups. An application to the problem of estimating the covolume of a discrete group of orientation-preserving isometries of hyperbolic 6-space is also given. This work was supported by Polish grant nr.0627/P3/93/04     

Topologically equivalent N-dimensional isometries

Let G be the group of isometries of the n-sphere, Euclidean n-space, or hyperbolic n-space, the group of similarities of Euclidean n-space, or the group of Möbius transformations of the n-sphere. In each case we attempt to determine the conjugacy classes in G which are amalgamated when we allow conjugation of the elements of G by homeomorphisms of the space on which G acts. We are successful modulo undetermined amalgamation among certain periodic orthogonal transformations.     

A Parametrization of Isometric Immersions between Hyperbolic Spaces

We parametrize the space of isometric immersions of the hyperbolic n -space into the hyperbolic ( n +1)-space by a family of properly chosen (at most) countable n -tuples of real-valued functions. This is an answer to the problem posed by Nomizu in 1973. We also construct a one-parameter family of deformable isometric immersions from the quotient spaces of the hyperbolic 2-plane modulo and some totally discontinuous subgroups of isometries.     

Standard position for objects in hyperbolic space

It is shown that, by applying hyperbolic isometries, one may arrangen points in hyperbolic (n–1)-space in a certain standard position. Similar results are developed for complex hyperbolic space, as well as hyperbolic spaces defined over nearly arbitrary fields. The algebraic basis of the paper is the determination of the structure of a double coset space which occurs in representation theory.Partially supported by NSA grant # MDA-904-96-0018.Partially supported by NSA grant # MDA904-95-1-1089.     

Dynamics of the Action of a CAT(0) Group on the Boundary

The main theorem here gives a geometric condition on the fixed point sets of two hyperbolic isometries of a CAT(0) group which guarantees that the subgroup generated by the two elements contains a free subgroup. A result in section 3 shows that an element extends to the identity on the boundary if and only if the element is virtually central in the CAT(0) group. Finally, the two dimensional case is considered in the last section of the paper and there it is shown that powers of two hyperbolic isometries commute if and only if a geometric condition on the fixed point sets is satisfied.     

Dualities and envelopes of one-parameter families of frontals in hyperbolic and de Sitter 2-spaces

We consider envelopes of one-parameter families of frontals in hyperbolic and de Sitter 2-space from the viewpoint of duality, respectively. Since the classical notions of envelopes for singular curves do not work, we have to find a new method to define the envelope for singular curves in hyperbolic space or de Sitter space. To do that, we first introduce notions of one-parameter families of Legendrian curves by using the Legendrian dualities. Afterwards, we give definitions of envelopes for the one-parameter families of frontals in hyperbolic and de Sitter 2-space, respectively. We investigate properties of the envelopes. At last, we give relationships among those envelopes.     

The Cannon–Thurston map for punctured-surface groups

Let Γ be the fundamental group of a compact surface group with non-empty boundary. We suppose that Γ admits a properly discontinuous strictly type preserving action on hyperbolic 3-space such that there is a positive lower bound on the translation lengths of loxodromic elements. We describe the Cannon–Thurston map in this case. In particular, we show that there is a continuous equivariant map of the circle to the boundary of hyperbolic 3-space, where the action on the circle is obtained by taking any finite-area complete hyperbolic structure on the surface, and lifting to the boundary of hyperbolic 2-space. We deduce that the limit set is locally connected, hence a dentrite in the singly degenerate case. Moreover, we show that the Cannon–Thurston map can be described topologically as the quotient of the circle by the equivalence relations arising from the ends of the quotient 3-manifold. For closed surface bundles over the circle, this was obtained by Cannon and Thurston. Some generalisations and variations have been obtained by Minsky, Mitra, Alperin, Dicks, Porti, McMullen and Cannon. We deduce that a finitely generated kleinian group with a positive lower bound on the translation lengths of loxodromics has a locally connected limit set assuming it is connected.     

The Lower Algebraic K -theory of Γ3

We explicitly compute the lower algebraic K-theory of Γ₃ a discrete subgroup of the group of isometries of hyperbolic 3-space. An erratum to this article is available at .     

关于小伸缩商拟共形群的几个定理

本文研究了单位球Bn上小伸缩商拟共形群的离散性质，给出了一个收敛定理，并且证明了在一定限制条件下任意一个非初等非离散小伸缩商拟共形群含有一个二元生成的非初等非离散子群。     

Fundamentals of Quaternionic Kinematics in Euclidean 4-Space

Based on a kinematic mapping for the group SE(4) of displacements of Euclidean 4-space, we show that the mapping of basic elements (points, oriented lines, oriented planes, oriented hyperplanes, instantaneous screws) can be written compactly in terms of 2 × 2 quaternionic matrices. Moreover we discuss the kinematics on the velocity level by investigating instantaneous screws and their geometric parameters.     

Collars in Complex and Quaternionic Hyperbolic Manifolds

We use the complex and quaternionic hyperbolic versions of Jørgensen's inequality to construct embedded collars about short, simple, closed geodesics in complex and quaternionic hyperbolic manifolds. In general, the width of these collars depend both on the length of the geodesic and on the rotational part of the group element uniformising it. For complex hyperbolic space we are able to use a lemma of Zagier to give an estimate based only on the length. We show that these canonical collars are disjoint from each other and from canonical cusps. We also calculate the volumes of these collars.     

Mean curvature one surfaces in hyperbolic space, and their relationship to minimal surfaces in Euclidean space

We describe local similarities and global differences between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. We also describe how to solve global period problems for constant mean curvature 1 surfaces in hyperbolic 3-space, and we give an overview of recent results on these surfaces. We include computer graphics of a number of examples.     

The Moduli Space of Complex Geodesics in the Complex Hyperbolic Plane

We consider the space M\mathcal{M} of ordered m-tuples of distinct complex geodesics in complex hyperbolic 2-space, H_\mathbbC²{\rm\bf H}_{\mathbb{C}}^{2}, up to its holomorphic isometry group PU(2,1). One of the important problems in complex hyperbolic geometry is to construct and describe the moduli space for M\mathcal{M}. This is motivated by the study of the deformation space of groups generated by reflections in complex geodesics. In the present paper, we give the complete solution to this problem.     

Cone metrics on the sphere and Livné’s lattices

We give an explicit construction of a family of lattices in PU (1, 2) originally constructed by Livné. Following Thurston, we construct these lattices as the modular group of certain Euclidean cone metrics on the sphere. We give connections between these groups and other groups of complex hyperbolic isometries.