Finite Groups of Mapping Classes of Surfaces
Finite Groups of Mapping Classes of Surfaces
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曲面映射类的有限群
DOI：
10.1007/bfb0090465
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发表时间：
1981
期刊：
影响因子：
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通讯作者：
H. Zieschang
中科院分区：
文献类型：
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作者：
H. Zieschang
这些笔记中心围绕尼尔森实现问题，即问题是否有限组映射类的表面可以实现有限组的映射。这是第一次回答积极的情况下，循环群中尼尔森使用几何参数开发的其他著名的学报论文。利用Rankke模，将[Nielsen 1942]中的定理推广到映射类的有限可解群的情形.这个假设是由于使用史密斯不动点定理。后来对一般情况给出了肯定的回答。在他的证明，他使用了一些曲率性质的Teichmüller空间这意味着一个更强的不动点定理比史密斯不动点定理。(The[Fenchel 1948，1950]的结果和方法在使用Teichmüller理论中被重新发现。根据[Kravetz 1959]的定理，对A. Karrass是否一个有限的torsionfree扩展的一个表面组再次是一个表面组;这个问题是有趣的，因为它是平行的类似问题的自由团体这是积极回答稍早在[Stallings 1968]和[天鹅1969]。然而，在同一时间在这里产生了一些怀疑的主要结果[Kravetz 1959年]和一个例子后来表明，结果Kravetz的曲率的Teichmüller空间是不正确的，因此，应用尼尔森实现问题是不一致的。另一方面，事实证明，正如一些例子所示，在[尼尔森1942]结束时的讨论是不正确的;然而，[尼尔森1942]的主要定理仍然有效，因为它是[Fenchel 1948，1950]的结果。尼尔森实现问题的答案一般是肯定的，但还没有详细的证明。部分答案，弱于那些可以得到的方法，
These Notes center around the Nielsen Realization Problem, ie the question whether a finite group of mapping classes of a surface can be realized by a finite groups of mappings. This was first answered positively for the case of cyclic groups in where Nielsen used geometrical arguments developed in the other famous Acta papers. Using Fricke moduli reproved and generalized the theorem of [Nielsen 1942] to the case of finite solvable groups of mapping classes. This assumption is due to the use of the Smith fixed point theorem. Later gave a positive answer for the general case. In his proof he used some curvature properties of Teichmüller spaces which imply a stronger fixed point theorem than the Smith fixed point theorem.(The result and approach of [Fenchel 1948, 1950] was rediscovered in using Teichmüller theory.) Based on the theorem of [Kravetz 1959] gave a positive answer to a question of A. Karrass whether a finite torsionfree extension of a surface group is again a surface group; this question was interesting because it is parallel to the similar problem on free groups which was answered positively slightly earlier in [Stallings 1968] and [Swan 1969]. However, at the same time here arose some doubts about the main result of [Kravetz 1959] and by an example later showed that the result of Kravetz on the curvature of the Teichmüller space is not true; hence, the application to the Nielsen Realization Problem is not consistently founded. On the other hand it turned out, as shown by some examples in that the discussion at the end of [Nielsen 1942] is not correct; nevertheless, the main theorem of [Nielsen 1942] remains valid, since it is a consequence of [Fenchel 1948, 1950]. That the answer to the Nielsen Realization Problem is positive in general has been claimed in a detailed proof is not yet available. Partial answers, weaker than those which can be obtained by the methods of