The Nielsen realization problem
The Nielsen realization problem
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尼尔森实现问题
DOI：
10.1090/s0273-0979-1980-14764-3
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发表时间：
1980
影响因子：
1.3
通讯作者：
S. Kerckhoff
中科院分区：
文献类型：
--
作者：
S. Kerckhoff
亏格g > 2的封闭定向曲面允许许多双曲（常高斯曲率-1）度量，这与高维中的Mostow刚性定理相反。只有特殊的双曲曲面有非平凡的等距群，但许多不同的，非同构群出现不同的对称度量。闭双曲曲面M2的等距群总是有限的，唯一的等距同位素的单位是单位本身。其结果是，双曲曲面与非平凡组的等距一直是一个主要来源的建设有限子群组的合痕类的M2，？TDiff（M2）。一个古老的问题，通常被称为尼尔森实现问题，是是否每个这样的有限子群产生作为一组等距的一些双曲曲面。在本文中，我们肯定地回答这个问题。
Closed, oriented surfaces of genus g > 2 admit many hyperbolic (constant Gaussian curvature -1) metrics in contrast to Mostow's rigidity theorems in higher dimensions. Only special hyperbolic surfaces have non-trivial groups of isometries, but many different, non-isomorphic groups arise for different symmetric metrics. The group of isometries of a closed hyperbolic surface M2 is always finite and the only isometry isotopic to the identity is the identity itself. As a result, hyperbolic surfaces with non-trivial groups of isometries have been a primary source for the construction of finite subgroups of the group of isotopy classes of diffeomorphisms of M2, ?TDiff(M2). An old question, usually referred to as the Nielsen Realization Problem, is whether every such finite subgroup arises as a group of isometries of some hyperbolic surface. In this paper we answer the question in the affirmative.