复射影结构的Schottky分解定理

具有Schottky完整性的复射影结构

Complex projective structures with Schottky holonomy

PSL（2，C）中的Schottky群诱导出一个开双曲曲面体，其理想边界是一个亏格等于Schottky群秩的闭可定向曲面S.该边界曲面具有（复）射影结构，其完整表示是从pi_1（S）到Schottky群的满射.我们将证明，一个任意的射影结构具有相同的holonomy表示是通过（2 π-）嫁接上述基本结构。

复射影结构与Schottky完整性的嫁接

Some Graftings of Complex Projective Structures with Schottky Holonomy

设$\mathcal{G}^*(S,ρ)$是一个图，它的顶点是具有完整性$ρ$的复投射结构，它的边是从一个顶点到另一个顶点的嫁接.如果$ρ$是拟Fuchsian的，则Goldman的定理暗示$\mathcal{G}^*(S,ρ)$是连通的。如果$ρ(π_1(S))$是一个肖特基群，巴巴已经证明了$\mathcal{G}(S,ρ)$（对应于未标记结构的图）是连通的。对于$ρ(π_1(S))$是Schottky基团的情况，本文给出了在基本情况下的接枝组成公式。使用这些公式，人们可以构建无限数量的（标准）投射结构，这些结构可以嫁接到一个公共结构上。此外，人们可以构造成对的投射结构，它们可以通过无数种方式嫁接连接。

Pentagon representations and complex projective structures on closed surfaces

We define a class of representations of the fundamental group of a closed surface of genus $2$ to $\mathrm{PSL}_2 (\mathbb C)$: the pentagon representations. We show that they are exactly the non-elementary $\mathrm{PSL}_2 (\mathbb C)$-representations of surface groups that do not admit a Schottky decomposition, i.e. a pants decomposition such that the restriction of the representation to each pair of pants is an isomorphism onto a Schottky group. In doing so, we exhibit a gap in the proof of Gallo, Kapovich and Marden that every non-elementary representation of a surface group to $\mathrm{PSL…

A structural description of extended ${\mathbb Z}_{2n}$-Schottky groups

Real points of Schottky space ${\mathcal S}_{g}$ are in correspondence with extended Kleinian groups $K$ containing, as a normal subgroup, a Schottky group $Γ$ of rank $g$ such that $K/Γ\cong {\mathbb Z}_{2n}$ for a suitable integer $n \geq 1$. These kind of groups are called extended ${\mathbb Z}_{2n}$-Schottky groups of rank $g$. In this paper, we provide a structural decomposition theorem, in terms of Klein-Maskit's combination theorems, of these kind of groups.

On Moduli Spaces of Convex Projective Structures on Surfaces: Outitude and Cell-Decomposition in Fock-Goncharov Coordinates

Generalising a seminal result of Epstein and Penner for cusped hyperbolic manifolds, Cooper and Long showed that each decorated strictly convex projective cusped manifold has a canonical cell decomposition. Penner used the former result to describe a natural cell decomposition of decorated Teichmüller space of punctured surfaces. We extend this cell decomposition to the moduli space of decorated strictly convex projective structures of finite volume on punctured surfaces. The proof uses Fock and Goncharov's $\mathcal{A}$-coordinates for doubly decorated structures. In addition, we describe a s…

尖点严格凸射影曲面的欧氏胞腔剖分算法

An algorithm for the Euclidean cell decomposition of a cusped strictly convex projective surface

Cooper和Long将Epstein和Penner的有限体积尖点双曲流形的欧氏胞腔剖分推广到有限体积的非紧严格凸射影流形。我们表明，周的算法来计算这种分解的双曲曲面推广到严格凸射影曲面。