Geometry of the complex of curves II: Hierarchical structure
Geometry of the complex of curves II: Hierarchical structure
复制标题
DOI：
10.1007/pl00001643
复制
发表时间：
1998-07
期刊：
影响因子：
--
通讯作者：
H. Masur;Y. Minsky
中科院分区：
文献类型：
--
作者：
H. Masur;Y. Minsky
本文继续对Harvey曲线复形进行几何研究，其最终目的是将双曲空间和群的理论应用于映射类群的算法问题和Kleian表示的几何性质。由于复数不是局部有限的，作者以前关于复数是增量双曲的结果很难应用，本文开发了一些工具来克服这个问题，并引入了一种组合机制，它描述了曲面上标记图中的基本运动序列。应用这些工具给出了映射类群中的一族拟测地字族，并给出了两个共轭伪Anosov元的最短字的一个线性界。分析中的一个基本工具是一族次表面投影，它大致类似于经典双曲空间中的钟球面的最近点投影。这些投影具有很强的收缩性质，这使得将复数的几何和作为顶点链接出现的(无限)子复形的几何联系在一起是可能的。由此产生的复合体的分层结构是通过一种称为测地线层次的组合装置来控制的，这是本文的中心结构。
This paper continues a geometric study of Harvey's Complex of Curves, whose ultimate goal is to apply the theory of hyperbolic spaces and groups to algorithmic questions for the Mapping Class Group and geometric properties of Kleinian representations. The authors' previous result that the complex is delta-hyperbolic was hard to apply because the complex is not locally finite; in this paper some tools are developed for overcoming this problem, and a combinatorial mechanism introduced which describes sequences of elementary moves in the graph of markings on a surface. These tools are applied to give a family of quasi-geodesic words in the Mapping Class Group, and a linear bound on the shortest word conjugating two conjugate pseudo-Anosov elements. A basic tool in the analysis is a family of subsurface projections, which are roughly analogous to closest-point projections to horoballs in classical hyperbolic space. These projections have a strong contraction property which makes it possible to tie together the geometry of the complex and that of the (infinite) subcomplexes that arise as links of vertices. The resulting layered structure of the complex is controlled by means of a combinatorial device called a hierarchy of geodesics, which is the central construction of the paper.