Various aspects of infinite groups of transformations acting on manifolds

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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