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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

自流形的概念提出以来，作用于流形上的变换群一直是人们研究的热点。在这项研究中，我们研究了变换的无限群，即在流形上光滑作用的无限群。在这一课题中，还有许多问题没有解决。我们研究的目的是研究无限变换群的研究方法，发展新的方法并应用于解决问题。（1）对于2维或3维流形的基本群等可表示的无限群的作用，阐明了作用的性质与几何群论中的量之间的关系。阐明了群的作用与群的有界上同调之间的关系。(2)阐明了各种无限变换群的分类空间的拓扑结构和无限变换群的不变量。在平价 ...更多信息在此，我们来看看辛同构或接触同构群的分类空间的拓扑。(3)我们研究无限变换群的动力学，定义与变换群的不变集相关的不变量，并研究不变量的性质。我们还研究了变换群的遍历性。(4)对流形上几个重要的无穷群作用进行了分类。特别是，我们期待在李群的离散子群流形上的行动的刚性。我们还研究了曲面映射类群的复解析作用的不变量。(5)我们研究了无穷群的几何学、无穷群的分类空间、无穷变换群的动力学、作用的刚性.我们保持研究人员之间的网络，以促进互动和合作。我们计划及时的会议，派遣研究人员到其他研究所，在国际会议上发表演讲，请外国专家审查我们的研究。在研究人员的合作下，我们从全球的角度进行了研究。这些研究都取得了成功，并在上面列出的路线中取得了自己的成果。特别是在日本数学学会会议上的5次特邀演讲（包括2次日本数学学会的几何奖）显示了我们的高研究水平。通过这些研究活动，下一个项目的新发展方向变得清晰。少