二元生成元Kleinian群的参数空间的二维切片

双生成元Kleinian轨道褶皱

Two-generator Kleinian orbifolds

本文给出了PSL（2，C）中不含不变平面且其生成元和换位子具有实迹的离散非初等两生成元子群所一致化的轨道折叠的完整列表。

Polynomial Trace Identities in $SL(2,{\bf C})$, Quaternion Algebras, and Two-generator Kleinian Groups

We study certain polynomial trace identities in the group $SL(2,\IC)$ and their application in the theory of discrete groups. We obtain canonical representations for two generator groups in §4 and then in §5 we give a new proof for Gehring and Martin's polynomial trace identities for good words, and extend that result to a larger class which is also closed under a semigroup operation inducing polynomial composition. This new approach is through the use of quaternion algebras over indefinites and an associated group of units. We obtain structure theorems for these quaternion algebras which appe…

On Gehring-Martin-Tan groups with an elliptic generator

The Gehring-Martin-Tan inequality for 2-generator subgroups of PSL(2,C) is one of the best known discreteness conditions. A Kleinian group $G$ is called a Gehring-Martin-Tan group if the equality holds for the group $G$. We give a method for constructing Gehring-Martin-Tan groups with a generator of order four and present some examples. These groups arise as groups of finite volume hyperbolic 3-orbifolds.

通过自连接研究Kleinian群的刚性

Rigidity of Kleinian groups via self-joinings

本文通过引入自连接高阶动力学的新视角，证明了在极限集上具有共形边界映射的非Fuchsian Kleinian群的新刚性定理，并回答了McMullen关于零体积四面体顶点映射刚性的问题。

信息词与离散性

Informative Words and Discreteness

在双曲三流形和Kleinian群与Fuchsian群的Teichm{ü}ller理论中，以及在两个生成矩阵群的离散问题中，经常会出现某些字族和字序列族（在两个生成矩阵群的生成元中的字）。我们调查的一些家庭这样的字和序列：半群的所谓的{ \sl好}字的Gehring-Martin，所谓的{ \sl杀手}字的Gabai-Meyerhoff-NThurston，Farey字的基恩系列和明斯基，离散算法Fibonacci序列的Gilman-Jiang和{ \sl抛物尘}字。我们调查家庭之间的联系，并建立一个新的好词和法里词之间的联系。

双曲空间中的Poincaré平分线

Poincaré Bisectors in Hyperbolic Spaces

本文给出了在双曲二、三空间中构造Dirichlet基本区的平分线的显式公式。他们进行了比较，等距领域的建设中采用的福特域，并用于找到一个有限集的发电机的离散群体的有限covolume。应用于富克斯群、Kleinian群，包括Bianchi群，以及有限幂零群的整群环的单位群的生成元的有限集的构造。给出了一个易于实现的算法DAFC，并将其用于离散群生成元的搜索。