On the geometry and dynamics of diffeomorphisms of surfaces
On the geometry and dynamics of diffeomorphisms of surfaces
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DOI：
10.1090/s0273-0979-1988-15685-6
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发表时间：
1988-10
影响因子：
1.3
通讯作者：
W. Thurston
中科院分区：
文献类型：
--
作者：
W. Thurston
大约12年前，这篇文章作为预印本广泛传播。当时《公报》不接受研究公告，在几次尝试出版后，我放弃了，预印本找不到归宿。我很快就发现了这一理论的许多后果，我在许多地方广泛地谈到了它。有一年，我的研究生课程专门研究这一理论，比尔·弗洛伊德和迈克尔·汉德尔在那门课上的笔记流传了一段时间。1976-1977年在奥赛举行的一次研讨会上，与会者仔细阅读了这些材料，并撰写了一卷[FLP]，其中也包括一些原始材料。从不同的角度来看，另一个很好的一般性参考是A.Casson的一套讲课笔记，由S.Bleler[CasBlee]记下。到目前为止，有几种可供选择的方法来发展这里描述的曲面的微分同态的分类。当我最初发现曲面的微分同胚的分类时，我不熟悉两个相当相关的数学体系：第一，黎曼曲面、拟共形映射和Teichmiiller理论；第二，尼尔森关于无限远处曲面的动力学行为的理论，以及他对测地线分层的近乎理解。Lipman Bers[Bersl]从被测叶片空间的角度听说了曲面自同构的分类，从Teichmüler理论的角度发展了曲面自同构分类的证明，通过允许Riemann曲面和映射的变化来推广Teichmiiller定理。丹尼斯·沙利文首先告诉我尼尔森的一些被忽视的文章，这些文章可能是相关的。R·米勒、J·吉尔曼、M·汉德尔和我都讨论过这个观点。三维测量层板和二维铁轨的相似理论已经有了长足的发展。这已经被应用于重新解释Haken的一些工作，在我、Hatcher、Floyd、Oertel等人的论文中以不同的组合将不可压缩曲面分类为特定的三维流形类。为了定义和分析SL(2，C)和SO(n，1)中的群的表示空间的紧性，Shalen，Morgan，Culler等人发展了作用于树上的群的相关理论，以及它与可测分层的关系，这有许多有趣的应用，包括三维流形中的不可压缩曲面理论。
This article was widely circulated as a preprint, about 12 years ago. At that time the Bulletin did not accept research announcements, and after a couple of attempts to publish it, I gave up, and the preprint did not find a home. I very soon saw that there were many ramifications of this theory, and I talked extensively about it in a number of places. One year I devoted my graduate course to this theory, and notes of Bill Floyd and Michael Handel from that course were circulated for a while. The participants in a seminar at Orsay in 1976-1977 went over this material, and wrote a volume [FLP] including some original material as well. Another good general reference, from a somewhat different point of view, is a set of notes of lectures by A. Casson, taken by S. Bleiler [CasBlei]. There are by now several alternative ways to develop the classification of diffeomorphisms of surfaces described here. At the time I originally discovered the classification of diffeomorphism of surfaces, I was unfamiliar with two bodies of mathematics which were quite relevant: first, Riemann surfaces, quasiconformal maps and Teichmiiller's theory; and second, Nielsen's theory of the dynamical behavior of surface at infinity, and his near-understanding of geodesic laminations. After hearing about the classification of surface automorphisms from the point of view of the space of measured foliations, Lipman Bers [Bersl] developed a proof of the classification of surface automorphisms from the point of view of Teichmüller theory, generalizing Teichmiiller's theorem by allowing the Riemann surface to vary as well as the map. Dennis Sullivan first told me of some neglected articles by Nielsen which might be relevant. This point of view has been discussed by R. Miller, J. Gilman, M. Handel and me. The analogous theory, of measured laminations and 2-dimensional train tracks in three dimensions, has been considerable development. This has been applied to reinterpret some of Haken's work, to classify incompressible surfaces in particular classes of 3-manifolds in papers by me, Hatcher, Floyd, Oertel and others in various combinations. Shalen, Morgan, Culler and others have developed the related theory of groups acting on trees, and its relation to measured laminations, to define and analyze compactifications of representation spaces of groups in SL(2, C) and SO(n, 1); this has many interesting applications, including the theory of incompressible surfaces in 3-manifolds.