中山大学数学系,Department of Mathematical Sciences Morehou8e College,Atlanta,GA 30314,USA,,广州 510275
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A Resolution of Some Mapping Classes
Li Xin LIU(1); Chao Hui ZHANG(
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Li Xin LIU Department of Mathematics,Zhongshan University,Guangzhou 510275,P.R.Chins Chao Hui ZHANG Department of Mathematical Sciences,Morehouse College,Atlanta,GA 30314,USA
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.
Li Xin LIU(1); Chao Hui ZHANG(.
A Resolution of Some Mapping Classes. Acta Mathematica Sinica, Chinese Series, 2007, 50(5): 1141-115 https://doi.org/10.12386/A2007sxxb0134