Synthetic studies of foliations and discrete group actions

这个项目的目的是研究叶理的几何和动力学性质，或者更一般地说，离散群作用。在整个项目中，我们得到了以下结果：1.当可解李群G作用在闭流形M上时，它决定了轨道的叶理。设G在M上的另一个作用具有相同的轨道叶状。我们考虑这两个作用是否C^∞共轭到G的一个同构的问题。我们用轨道叶理的叶理上同调解释了这个问题，并给出了一些作用的例子，证明了这个问题的正解。2.证明了闭区间上的一些保意义复同态群是完美的。例子有Lipschitz同胚群、端点与恒等式相切的C ^∞同胚群、端点与恒等式相切的C^1同胚群。3.设闭3-流形上的两个余维1叶理横切。是否有可能对一个叶理进行同位素测定，使它也与另一个叶理相切，但方式不同？我们研究了这个问题，证明了双曲环面自同构的悬浮流的稳定和不稳定叶理具有唯一的相交性质，而双曲曲面的测地线流的稳定和不稳定叶理则没有相交性质。

The purpose of this project is to study the geometic and dynamical properties of foliations, or more generally, of discrete group actions. Throughout the project, we have obtained the following results.1.When a solvable Lie group G acts on a closed manifold M, it determines the orbit foliation. Assume another G action on M with the same orbit foliation is given. We consider the problem whether or not the two actions are C^∞ conjugate up to an isomorphism of G. We interpreted this problem in terms of the foliated cohomology of the orbit foliations, and gave some examples of actions for which this problem has a positive answer.2.Some groups of the sense preserving diffeomorphisms of the closed interval are shown to be perfect. The examples are, the group of Lipschitz homeomorphism, the group of C^∞ diffeomorphisms which are C^∞ tangent to the identity at the end points, and the group of C^1 diffeomorphisms C^1 tangent to the identity at the end points.3.Suppose two codimension one foliations on a closed 3-manifold intersect transversely. Can it possible to isotope one foliation so that it is also tangent to the other, but in a different way? We considered this problem and have shown that the stable and unstable foliations of the suspension flow of hyperbolic toral automorphisms have unique intersection property, but those of geodesic flows of hyperbolic surfaces have not.