The Pants Complex Has Only One End
The Pants Complex Has Only One End
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裤子复合体只有一个终点
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
S. Schleimer
中科院分区：
文献类型：
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作者：
H. Masur;S. Schleimer
回想一下，S的Pants分解由S上的3g（S）−3个不相交的本质非平行简单闭曲线组成。曲线补数的每个分量是具有3个孔的球体或一条裤子。则裤复形CP（S）是度量图，其顶点是S的裤分解，直到合痕。两个顶点P，P ′通过一条边相连，如果P，P ′相差一个初等移动.在基本移动中，裤子中的所有曲线都是固定的，除了一条曲线α。除去α，剩余曲线的补曲线中包含α的分量要么是一个穿孔环面，要么是一个有4个孔的球面。然后α被包含在这个区域中的曲线β所取代，该曲线与α相交最小;在穿孔环面的情况下一次，在球体的情况下两次。CP（S）的所有边都被分配长度1。设d（·，·）为CP（S）中的距离函数。已知裤子复形CP（S）是连通的[HT]。回想一下，度量空间（X，d）有一个端点，如果对于任何基点O ∈ X和任何半径R，BR = BR（O）的补，即半径R以O为中心的球，只有一个无界分支。很容易看出，定义并不依赖于点O的选择。显然有一端是一个准等距不变量。因此，遵循Brock [B]，我们的定理意味着：
Recall that a pants decomposition of S consists of 3g(S)−3 disjoint essential non-parallel simple closed curves on S. Each component of the complement of the curves is a sphere with 3 holes or pair of pants. Then the pants complex CP (S) is the metric graph whose vertices are pants decompositions of S, up to isotopy. Two vertices P,P ′ are connected by an edge if P,P ′ differ by an elementary move. In an elementary move all curves in the pants are fixed except for one curve α. Removing α, the component of the complement of the remaining curves that contains α is either a punctured torus or a sphere with 4 holes. Then α is replaced by a curve β contained in this domain that intersects α minimally; in the punctured torus case once, and in the sphere case twice. All edges of CP (S) are assigned length 1. We let d(·, ·) be the distance function in CP (S). The pants complex CP (S) is known to be connected [HT]. Recall that a metric space (X, d) has one end if for any basepoint O ∈ X and any radius R the complement of BR = BR(O), the ball of radius R centered at O, has only one unbounded component. It is easy to see that the definition does not depend on the choice of the point O. Clearly having one end is a quasi-isometry invariant. So, following Brock [B], our theorem implies: