Various aspects of infinite groups of transformations acting on manifolds

自流形概念建立以来，作用于流形的变换群已经被深入研究。在本研究中，我们研究无穷变换群，即无穷群在流形上的光滑作用。在这门学科中，还有许多尚未解决的问题。我们的研究目的是研究无穷群变换的研究技术，开发新的技术并将其应用于解决问题。(1)关于有限呈现的无限群的作用，如二维或三维流形的基本群，我们阐明了作用的性质与几何群论中量的关系。澄清了群的作用与群的有界上同调之间的关系。(2)澄清了各种无穷变换群的分类空间拓扑和无穷变换群的不变量。更具体地说，我们研究了辛形态群或接触形态群的分类空间的拓扑结构。(3)研究了无穷变换群的动力学，定义了变换群的不变量与不变量集的关联，研究了不变量的性质。我们还研究了变换群的遍历性质。(4)我们分类了流形上几个重要的无限群作用。特别地，我们研究李群的离散子群流形上作用的刚性。我们还研究了曲面的映射类群的复解析作用的不变量。(5)讨论了无限群的几何研究、无限群的分类空间研究、无限变换群的动力学研究、作用的刚性研究等之间的关系，并对纤维束的束变换群、曲面的保面积微分同构群、复流形的复解析变换群澄清了这种关系。为了进行这些研究，我们维持了研究人员之间的网络，以促进互动和合作。我们及时安排会议，派遣研究人员到其他研究所，在国际会议上做报告，请外国专家来审查我们的研究。在研究人员的合作下，我们从全球的角度进行了这项研究。这些研究成功地完成了，他们自己的结果在上面列出的行。特别是在日本数学会会议上的5次特邀演讲（其中2次获得日本数学会几何奖）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起来。少

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