一类曲面映射类的分解  

A Resolution of Some Mapping Classes

作　　者：刘立新[1] 张朝晖 

机构地区：[1]中山大学数学系 [2]Department of Mathematical Sciences,Morehou8e College,Atlanta,GA 30314,USA

出　　处：《数学学报（中文版）》2007年第5期1141-1150,共10页Acta Mathematica Sinica：Chinese Series

摘　　要：设S为至少有一个穿孔点α的Riemann曲面.对于曲线α■S,可以定义关于α■S的Dehn twist t_α.设H是S的映射类群的子群,H中的元素保持α不动,并且投影为S=S∪{α}上平凡的映射类变换.定义t_α是关于α■S的Dehn twist.本文考虑关于X(S上的映射类变换)的方程(t_α■θ)~n■X=圮,其中θ∈H是任意给定的.由于(t_αoθ)~n和t_α~n都投影为关于简单闭曲线■的Dehn twist t_■.所以上述方程在H中的解是存在的.对充分大的n,我们给出上述方程有形如X=θ^(n)的解的充要条件.此外,对任给的θ∈H,刻画了子空间H′■H,这里方程的解X=X_n最终要属于H′.最后,考虑简单映射类变换的某些复合映射,并给出了相应的刻画:它们在沿S上的某些简单曲线做剖分后所得的穿孔pant上是不可约的.Let S be a Riemann surface with at least one puncture a. There is a group H of mapping classes on S that fixes a and projects to the trivial mapping classes of S=S∪{α}. Let tα denote the Dehn twist along a simple closed curve α on S. In this paper we study the equation of mapping classes of S for X： （tαοθ）^nοX=tα^n, where θ ∈ H is given. Since both （tαοθ）^n and tα^n project to the same Dehn twist t5 along a simple loop αbelong to S, the above equation always has a solution in H. We give the necessary and sufficient condition for the equation to have a solution X =θ^-n for all sufficiently large n. We also characterize for an arbitrarily given θ∈ H a subset H＇ C H such that the solution X =Xn for the equation eventually lies in H＇. Finally, we study some compositions of simple mapping class and characterize that they are irreducible on a punctured pair of pants obtained from S by cutting along some simple curvcs.

关 键 词：TEICHMÜLLER空间 映射类 Bers纤维空间 

分 类 号：O174.5[理学—数学]