生成代数映射类群的Zieschang-McCool方法
The Zieschang-McCool method for generating algebraic mapping-class groups
摘要：
设g和p为非负整数。设A（g，p）表示由{t_1，.上的自由群的所有自同构组成的群，t_p，x_1，.，x_g，y_1，. y_g}，其固定元素t_1t_2. t_p[x_1，y_1]. [x_g，y_g]并置换共轭类的集合{[t_1]，.，[t_p]}。Labruère和巴黎，建立在工作的阿廷，马格努斯，德恩，尼尔森，Lickorish，Zieschang，Birman，汉弗莱斯，和其他人，表明，A（g，p）是由一个集，这是所谓的ADLH集。我们使用的方法Zieschang和McCool给一个自足的，代数证明这一结果。Labruère和巴黎也给出了定义关系的ADLH集在A（g，p）;我们不知道一个代数证明这g > 1。考虑亏格为g的可定向曲面S（g，p），有p个穿孔，使得（g，p）不是（0，0）或（0，1）. S（g，p）的代数映射类群记作M（g，p），定义为具有生成集{t_1，.，t_p，x_1，.，x_g，y_1，. y_g}和关系式t_1t_2. t_p[x_1，y_1]. [x_g，y_g]，其置换共轭类的集合{[t_1]，.，[t_p]}。现在从尼尔森的结果得出，M（g，p）是由ADLH集的图像与反射一起生成的。这给出了一种新的方法来证明M（g，p）等于S（g，p）的（拓扑）映射类群，沿着Magnus、Karrass和Solitar在1966年提出的路线。
英文摘要：
Let g and p be non-negative integers. Let A(g,p) denote the group consisting of all those automorphisms of the free group on {t_1,...,t_p, x_1,...,x_g, y_1,...y_g} which fix the element t_1t_2...t_p[x_1,y_1]...[x_g,y_g] and permute the set of conjugacy classes {[t_1],....,[t_p]}. Labruère and Paris, building on work of Artin, Magnus, Dehn, Nielsen, Lickorish, Zieschang, Birman, Humphries, and others, showed that A(g,p) is generated by a set that is called the ADLH set. We use methods of Zieschang and McCool to give a self-contained, algebraic proof of this result. Labruère and Paris also gave defining relations for the ADLH set in A(g,p); we do not know an algebraic proof of this for g > 1. Consider an orientable surface S(g,p) of genus g with p punctures, such that (g,p) is not (0,0) or (0,1). The algebraic mapping-class group of S(g,p), denoted M(g,p), is defined as the group of all those outer automorphisms of the one-relator group with generating set {t_1,...,t_p, x_1,...,x_g, y_1,...y_g} and relator t_1t_2...t_p[x_1,y_1]...[x_g,y_g] which permute the set of conjugacy classes {[t_1],....,[t_p]}. It now follows from a result of Nielsen that M(g,p) is generated by the image of the ADLH set together with a reflection. This gives a new way of seeing that M(g,p) equals the (topological) mapping-class group of S(g,p), along lines suggested by Magnus, Karrass, and Solitar in 1966.
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