可交换的映射类及其在无穷远边界上的作用

Commuting Mapping Classes and Their Actions on the Circle at Infinity

设■为亏格p≥2的紧致Riemann曲面,a为■上的任意一点.S=■-a.我们知道,映射类群Mods中的任一元素θ保持穿孔点a不动,因而θ可投影为映射类群Mod_s中的一元素.根据Bers,θ等同于一个单位圆盘D上的一个拟共形映射■的等价类[■],它由■的边界值唯一确定.本文考虑映射类群Mod_S中的一些元素的混合,这些元素要么是简单的Dehn twists,要么是那些投影为■上恒等的映射类,通过复合这些不同类型的元素,并且考察它们在单位圆边界上的迭代,给出一些等价条件使得它们生成映射类群Mod_S的阿贝尔子群.

Let S be a compact Riemann surface of genus p ≥ 2. Let a be a point on S and let S = S／{α}. We know that every element θ in the mapping class group Mods of S fixes the puncture α, and thus θ can be projected to an element of the mapping class group Mode. According to Bers, the element θ is identified with an equivalence class [f] of a quasiconformal self-map f of the unit disk D that is determined by the boundary value f｜s1. In this paper, we consider some special elements of Mods that are either simple Dehn twists or the mapping classes projecting to the identity on S. By mixing elements with different types and by investigating their iterations on the circle at infinity, we give some equivalent conditions for those elements to generate an Abelian subgroup of Mods.