一类曲面映射类的分解

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.

A Resolution of Some Mapping Classes

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 ■=SU{a}.Let t_αdenote the Dehn twist along a simple closed curveαon S. In this paper we study the equation of mapping classes of S for X:(t_αoθ)~n o X= t_■~n,whereθ∈H is given.Since both (t_αoθ)~n and t_α~n project to the same Dehn twist t_■ along a simple loop ■,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 givenθ∈H a subset H′■ H such that the solution X=X_n 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 curves.