Pleating coordinates for the Maskit embedding of the Teichmuller space of punctured tori
Pleating coordinates for the Maskit embedding of the Teichmuller space of punctured tori
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穿孔环面 Teichmuller 空间的 Maskit 嵌入的打褶坐标
DOI：
10.1016/0040-9383(93)90048-z
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发表时间：
1993
期刊：
影响因子：
--
通讯作者：
C. Series
中科院分区：
文献类型：
--
作者：
L. Keen;C. Series
在本文中，我们引入了一组新的参数，我们称之为破环体的Teichmiiller空间Tr， 1的褶坐标。坐标网格如图1所示。当t1, 1作为Kleinian群{G，}的全纯族嵌入时，这些坐标具有这样的几何构型，这些全纯族依赖于在c的单连通域4中变化的复参数u。嵌入的方式是使正则集R （G,,）具有唯一不变分量R，（G,），并且Tr， R中的点由黎曼曲面QO (GP)/G，表示。这种嵌入称为（T1, 1）的掩码嵌入。（请参阅第2节，在这里获得对技术术语和思想的更轻松和详细的解释。）我们的坐标有三个优点：首先，它们直接关系到双曲流形H3/G的几何形状，或者更准确地说，关系到凸壳边界“面向”& （G,）的分量He（见第4.2节）；其次，它们完全反映了人们在极限集中看到的视觉模式；第三，它们可以直接从GP的生成器中计算。如图1所示，可以很清楚地看到JZ是如何位于c内部的。在Gfl下，凸壳的边界是不变的，并且可以从被刺穿的环面的几何形状$= aq /GP中读取坐标。表面a%带有自然双曲度规，沿测地线折叠，投射到gP上的测地线层合d。网格中的“垂直”线表示1保持固定的线。我们称这样的线为褶线。穿孔环面上所有可能的层合的集合自然被识别为fi，除了与cc对应的层合外，所有层合都确定褶皱射线。在图1中，射线沿r按自然顺序出现。对于1 Er，褶皱射线PA渐近于实线' 33~= 22为p+ co。1 EQ层压是gP上的简单封闭测地线y (n)。如果gn (p) EG,，是表示r （L)的元素，则射线~ 3′~与轨迹{p EC: Tr gl (p)> 2）的唯一分支重合。这些有理射线在A中密度很大，根据McMullen[19]最近的结果，它们的端点在A &中密度很大。沿着有理射线，R,（G,）是重叠圆的并集，它们以一种反映a的连分式展开的方式组合在一起。这些模式在视觉上是明显的，至少对于d&Z附近的~ 1值，在这些群的极限集的图片中是如此，如图2所示。我们对这些模式的兴趣是David Wright在对8&进行计算机调查的过程中发现的，这是我们在这里工作的最初动机。
IN THIS paper we introduce a new set of parameters that we call pleating coordinates for the Teichmiiller space Tr, 1 of the punctured torus. The coordinate grid is shown in Fig. 1. These coordinates have this geometric configuration when T 1, 1 is embedded as a holomorphic family of Kleinian groups {G,} depending on a complex parameter, u that varies in a simply connected domain 4 in C. The embedding is made in such a way that the regular set R (G,,) has a unique invariant component R,(G,) and the points in Tr, r are represented by the Riemann surfaces QO (GP)/G,. This embedding is known as the Maskit embedding for T1, 1.(See Section 2 for a more leisurely and detailed explanation of the technical terms and ideas here.)The advantages of our coordinates are threefold: first, they relate directly to the geometry of the hyperbolic manifold H3/G,, or more precisely to the component He of the convex hull boundary “facing” & (G,)(see Section 4.2); second, they reflect exactly the visual patterns one sees in the limit sets; and third, they are directly computable from the generators of GP. As is apparent in Fig. 1, one sees quite explicitly how JZ sits inside C. The boundary of the convex hull is invariant under Gfl and the coordinates can be read off from the geometry of the punctured torus $= aqO/GP. The surface a%,, carries a natural hyperbolic metric and is pleated along geodesics that project to a geodesic lamination d on gP. The “vertical” lines in the grid represent lines along which 1 remains fixed. We call such a line a pleating ray. The set of all possible laminations on a punctured torus is naturally identified with fi and all the laminations except the one corresponding to cc determine pleating rays. The rays appear in Fig. 1 in their natural order along R. For 1 Er, the pleating ray PA is asymptotic to the real line ‘33~= 22 as p+ co. When; 1 EQ the lamination is a simple closed geodesic y (n) on gP. If gn (p) EG,, is an element representing r (L), the ray~ 3’~ coincides with a unique branch of the locus {p EC: Tr gl (p)> 2). These rational rays are dense in A, and by a recent result of McMullen [19], their endpoints are dense in a&. Along the rational rays, R,(G,) is a union of overlapping circles that fit together in a manner reflecting the continued fraction expansion of A. These patterns are visually apparent, at least for values of~ 1 near d&Z, in pictures of the limit sets of these groups as we see in Fig. 2. Our interest in these patterns, discovered by David Wright in the course of a computer investigation of 8&, was the original motivation for the work here.