Various aspects of infinite groups of transformations acting on manifolds

作用于流形上的无限变换群的各个方面

基本信息

批准号：
16204004
负责人：
TSUBOI Takashi
金额：
$ 23.13万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16204004/
关键词：
infinite groups transformation grouns geometric theory of groups diffeomorphism groups classifying spaces characteristic classes dynamics of group actions invariant sets 幾何学多様体無限変換群葉層構造曲面束幾何的群論横断構造接触構造クライン群極小集合 /

项目摘要

Transformation groups acting on manifolds have been intensively investigated since the establishment of the notion of manifolds. In this research, we look at infinite groups of transformations, that is, infinite groups acting smoothly on manifolds. In this subject, there are still many unsolved problems. The aim of our research was to study the research techniques on infinite groups of transformations, to develop new techniques and to apply it to solve the problems. More specifically, it was stated as follows (1) On the action of finitely presented infinite groups such as the fundamental groups of 2 or 3 dimensional manifolds, we clarify the relationship between the nature of the actions and the quantities coming from geometric group theory. We clarify the relation between the action of groups and the bounded cohomology of groups. (2) We clarify the topology of the classifying spaces of various infinite transformation groups and the invariants for infinite transformation groups. In par … More ticular, we look at the topology of the classifying space for the group of symplectomorphisms or contactomorphisms. (3) We study the dynamics of infinite transformation groups, define invariants associated with invariant sets for the transformation groups and study the properties of the invariants. We also study the ergodic properties of transformation groups. (4) We classify several important infinite group actions on manifolds. In particular, we look at the rigidity of the actions on manifolds of discrete subgroups of Lie groups. We also investigate the invariants of complex analytic actions of mapping class groups of surfaces. (5) We look at the relationship among the study on the geometry of infinite groups, on classifying spaces of infinite groups, on dynamics of infinite transformation groups, on rigidity of actions, … We clarify this relation for the bundle transformation groups of fiber bundles, the area preserving diffeomorphism groups of surfaces, the complex analytic transformation groupoid of complex manifolds.To perform these researches, we maintained the network among the researchers to promote the interaction and the collaborations. We planned timely meetings, sent researchers to other institute, gave presentations in the international meetings, asked being foreign specialists to review our research. With the collaboration of researchers, we performed the research from a global point of view.These researches were done successfully with their own results in the line listed above. In particular, our high research level was shown by 5 special invited lectures in -the meeting of the Mathematical Society of Japan (including 2 Geometry Prize of the Mathematical Society of Japan awards).By these research activities, the direction of new development for the next project became clear. Less

自建立多种多样的概念以来，已经对作用于多种流形的转型群体进行了深入的研究。在这项研究中，我们研究了无限的转换组，即无限群在流形上顺利作用。在这个主题中，仍然存在许多未解决的问题。我们研究的目的是研究无限转型群体的研究技术，开发新技术并应用其解决问题。更具体地说，如下（1）关于最终提出的无限群体（例如2或3维流形的基本组）的作用（1），我们阐明了行动的性质与来自几何组理论的数量之间的关系。我们阐明了群体的作用与群体有限的共同体之间的关系。（2）我们阐明了各种无限转化组的分类空间和无限转化组的不变空间的拓扑结构。 …更典型的是，我们研究了符号术或接触型的分类空间的拓扑结构。（3）我们研究无限转化组的动力学，即与变换组的不变集相关的定义不变性，并研究了不变性的特性。我们还研究了转化组的厄法德特性。（4）我们对流形的几个重要的无限群体行为进行了分类。特别是，我们探讨了谎言组离散亚组的歧管的僵化。我们还研究了映射表面类别组的复杂分析作用的不变性。（5）我们研究有关无限群体几何形状的研究，无限群体的分类，无限转化组的动态，行动的刚性，……我们阐明了纤维捆绑包的捆绑转换群的这种关系，该关系是维护了复杂的分析群体的范围的研究，我们维护了复杂的研究群体，我们维护了复杂的研究群体，我们宣布了复杂的分析群体，从而宣布了这些范围的研究。互动和协作。我们计划及时的会议，将研究人员派往其他学院，在国际会议上进行演讲，要求作为外国专家审查我们的研究。通过研究人员的合作，我们从全球的角度进行了研究。这些研究在上面列出的行中成功完成了自己的结果。特别是，在日本数学学会会议上的5个特殊邀请的讲座（包括日本数学学会的2几何奖）中，我们的研究水平提出了高度的研究水平。通过这些研究活动，下一个项目的新发展方向变得明确。较少的