SURFACES OF GENUS 2: GENERIC FUNDAMENTAL POLYGONS
SURFACES OF GENUS 2: GENERIC FUNDAMENTAL POLYGONS
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第 2 类曲面：通用基本多边形
DOI：
10.1093/qmath/33.4.451
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发表时间：
1982
影响因子：
0.7
通讯作者：
M. Näätänen
中科院分区：
文献类型：
--
作者：
T. Jørgensen;M. Näätänen
当研究作用于双曲平面H上的Fuchsian群F时，通常用一个基本多边形来表示轨道空间HfT是很方便的。构造这样一个多边形的方法有很多种。如果选择在F作用下具有不同图像的点h和h，则在h的F轨道上离h较近的点的集合Ph (F)是一个凸基本多边形。Ph (F)的边与F的某些元素成对全等，这些边对变换产生F。我们研究了当HfT是一个2属的闭曲面S时，类似Ph (F)的多边形的一些可能的边对模式。人们可以认为S是通过将基本多边形p的等价边粘合在一起而得到的。在表面上，粘合轨迹是一个连通补的连通图G。绕P的边界走一次对应于在G上的闭合行走，其中每条边在每个方向上都经过一次。设e和v表示g的边和顶点的数目，因为S有2属，欧拉公式给出e= v+ 3。G的每条边提升到P的2条边，每个顶点提升到至少3个顶点。P的边数等于顶点数，因此是2e^ 3D。由此可见，P最多有18条边。当v= 1时，P的最小边数为8。一般的情况是Ph (F)是一个18-gon，或者等价地，G是一个有9条边和6个顶点的三价图。给定群F，对于几乎所有h [1]， P= Ph (F)出现了这幅图。我们证明，在不考虑方向的情况下，有8种本质上不同的方法来配对一个18-gon的边，以获得一个2属的封闭曲面（定理1）。这是通过首先展示可能的图形（提案1），然后研究哪些封闭的路径是允许的（提案2）来实现的。第5节介绍了这8种类型之间的转换模式。所有8种模式都以狄利克雷区域的形式出现。用庞加莱定理构造一个角度为2TT/3的18形圆锥体可以看出这一点。在计算机的帮助下，Lee Mosher发现3属曲面有1726种不同类型的通用基本多边形。
WHEN studying a Fuchsian group F acting on the hyperbolic plane H it is often convenient to represent the orbit space HfT by a fundamental polygon. There are various ways of constructing such a polygon. If one chooses a point h e H with distinct images under the action of F, then the set Ph (F) of points closer to h than to any other point in the F-orbit of h is a convex fundamental polygon. The sides of Ph (F) are congruent in pairs by certain elements of F. These side-pairing transformations generate F.We study some of the different possible side-pairing patterns for polygons like Ph (F) in the case when HfT is a closed surface S of genus 2. One may think of S as being obtained by gluing together the equivalent sides of a fundamental polygon P. On the surface the gluing locus is a connected graph G with connected complement. Going around the boundary of P once corresponds to a closed walk on G, where each edge is traversed exactly once in each direction. Let e and v denote the number of edges and vertices of G. Since S has genus 2, Euler's formula gives e= v+ 3. Each edge of G lifts to 2 sides of P and each vertex lifts to at least 3 vertices. The number of sides of P equals the number of vertices, hence 2e^ 3D. It follows that P has at most 18 sides. The minimal number of sides for P is 8, obtained when v= 1. The generic case is that of Ph (F) being an 18-gon or, equivalently, of G being a trivalent graph with 9 edges and 6 vertices. Given the group F, this picture arises for P= Ph (F) for almost all h [1]. We show that, disregarding orientation, there are 8 essentially different ways of pairing the sides of an 18-gon to obtain a closed surface of genus 2 (Theorem 1). This is done by first exhibiting the possible graphs (Proposition 1) and then studying which closed walks each permits (Proposition 2). The transition pattern between the 8 types is presented in Section 5. All 8 patterns occur as Dirichlet regions. This is seen by constructing an 18-gon with angles 2TT/3 and using the Poincare theorem. With help of a computer, Lee Mosher has found that there are 1726 different types of generic fundamental polygons for surfaces of genus 3.