Classical Two-parabolic T-Schottky groups

A sufficiently complicated noded Schottky group of rank three

The theoretical existence of non-classical Schottky groups is due to Marden. Explicit examples of such kind of groups are only known in rank two, the first one by by Yamamoto in 1991 and later by Williams in 2009. In 2006, Maskit and the author provided a theoretical method to obtain examples of non-classical Schottky groups in any rank. The method assumes the knowledge of some algebraic limits of Schottky groups, called sufficiently complicated noded Schottky groups, whose existence was stated. In this paper we provide an explicit construction of a sufficiently complicated noded Schottky grou…

McShane's identity for classical Schottky Groups

Greg McShane introduced a remarkable identity for the lengths of simple closed geodesics on cusped hyperbolic surfaces. This was subsequently generalized by the authors to hyperbolic cone-surfaces, possibly with cusps and/or geodesic boundary. In this paper, we generalize the identity further to the case of classical Schottky groups. As a consequence, we obtain some surprising new identities in the case of fuchsian Schottky groups. For classical Schottky groups of rank 2, we also give generalizations of the Weierstrass identities, first given by McShane.

Kleinian groups of small Hausdorff dimension are classical Schottky groups I

In this paper we prove that there exists a positive number $λ>0$, such that any 2-generated Kleinian groups with limit set of Hausdorff dimension $<λ$ are classical Schottky groups.

经典 Schottky 群的收缩格扩张

Systolic lattice extensions of classical Schottky groups

本文构造了经典 Schottky 群的收缩格扩张，证明闭双曲流形收缩元实现的复平移长度具有稠密性，并给出构造非算术格的新方法。

关于具有三个抛物元素的二阶克莱因群

On rank two Kleinian groups with three parabolics

研究二阶自由群到\(\mathsf{PSL}(2,\mathbb{C})\)的离散表示，使三条闭曲线映为抛物元素，证明无限余体积群是特定极大尖点群，还发现了非来自特定链环希盖尔分裂表示的有限余体积群。

高维Schottky空间的拓扑

The topology of Schottky spaces in higher dimensions

研究高维Schottky空间的拓扑结构，主要定理表明在边界维数（两倍秩）时，空间的基本群是二阶循环群的乘积，但整体单连通，并证明旋转对称核是强形变收缩核。

The diagonal slice of Schottky space

An irreducible representation of the free group on two generators X,Y into SL(2,C) is determined up to conjugation by the traces of X,Y and XY. We study the diagonal slice of representations for which X,Y and XY have equal trace. Using the three-fold symmetry and Keen-Series pleating rays we locate those groups which are free and discrete, in which case the resulting hyperbolic manifold is a genus-2 handlebody. We also compute the Bowditch set, consisting of those representations for which no primitive elements in the group generated by X,Y are parabolic or elliptic, and at most finitely many …

无限亏格的把手体与肖特基群

Handlebodies of Infinite Genus and Schottky Groups

本文通过定义展开对无限亏格把手体、无限生成肖特基群及单值化问题的研究，证明相关曲面和把手体的拓扑单值化、特定黎曼曲面的拟共形单值化及唯一性，并给出反例。

The classification of Kleinian groups of Hausdorff dimensions at most one and Burnside's conjecture

In this paper we provide the complete classification of convex cocompact Kleinian group of Hausdorff dimensions less than $1.$ In particular, we prove that every convex cocompact Kleinian group of Hausdorff dimension $<1$ is a classical Schottky group. This upper bound is sharp. The result implies that the converse of Burside's conjecture \cite{Burside} is true: All non-classical Schottky groups must have Hausdorff dimension $\ge1$. The prove of the theorem relies on the result of Hou \cite{Hou}.

The classification of Kleinian groups of Hausdorff dimensions at most one

In this paper we provide the complete classification of Kleinian groups of Hausdorff dimensions less than $1.$ In particular, we prove that every purely loxodromic Kleinian groups of Hausdorff dimension $<1$ is a classical Schottky group. This upper bound is sharp. As an application, the result of \cite{H} then implies that, every closed Riemann surface is uniformizable by a classical Schottky group. The prove relie on the result of Hou \cite{Hou}, and space of rectifiable $\G$-invariant closed curves.

Riley 切片的逼近

Approximations of the Riley slice

本文借鉴 Keen 和 Series 的思想，显式确定了完全位于 Riley 切片内的褶皱射线“半空间”邻域，提供了一种可证明的判定方法，并通过 Farey 多项式族给出了切片的动力系统解释。

The Riley slice revisited

In [4]: `The Riley slice of Schottky space', (Proc. London Math. Soc. 69 (1994), 72-90), Keen and Series analysed the theory of pleating coordinates in the context of the Riley slice of Schottky space R, the deformation space of a genus two handlebody generated by two parabolics. This theory aims to give a complete description of the deformation space of a holomorphic family of Kleinian groups in terms of the bending lamination of the convex hull boundary of the associated three manifold. In this note, we review the present status of the theory and discuss more carefully than in [4] the enumer…