STUDY ON INFINITE TRANSFORMATION GROUPS ON MANIFOLDS

1.研究了群的有限生成和有限表示的条件。我们证明了群作用于单连通空间，使得商空间是有限单形，且迷向群是有限生成和有限表示的。我们还收集了大量关于有限生成和有限压力群的知识。对微分同胚群的分类空间的研究扩展到极限集或端集的同胚群的分类空间的研究。我们得到了关于它们的拓扑学的几个结果。特别地，我们证明了Menger紧空间的同胚群的分类空间是非循环的。我们于1995.3年在日本和国外公布了这一结果。我们研究了李群的离散子群。我们研究了保角变换群的极限集。我们构造了几个离散保角变换群的例子。研究它与数论的关系将是很有趣的。我们从动力系统的角度分析了保测变换。我们研究了动力系统的Calabi不变量和可微性。1996年，我们发现了保持圆盘的Arca可微同胚群的Calabi不变量与圆盘的可微同胚群的Euler类之间的关系。这在未来的调查中应该是重要的。我们构造了一些有趣的加压变换的例子。我们将继续对它们进行分析。我们问了教授。日内瓦大学的Hacfliger审查我们的调查。我们在调查和调查中得到了很高的分数，也得到了宝贵的建议。

1. We studied the condition for a group to be finitcly gencrated and finitely presented. We gave a proof for the case where the group acts on a simply connceted space such that the quotient space is a finite simplex and the isotropy groups are finitely gencrated and finitely presented. We also gathered a lot of knowledge on the finitely generated and finitely prescnted groups.2. The study of the classifying space for the group of diffeomorphisms extends to the study of the classifying spaces for the groups of homeomorphisms of the limit sets or the end sets. We obtained several results on the topology of them. In particular, we showed that the classifying space of the group of homeomorphisms of the Menger compact space is acyclic. We presented this result in Japan and on abroad in 1995.3. We studied discrete subgroups of Lie groups. We looked at the limit set for conformal transformation groups. We constructed several examples of discrete conformal transformation groups. It will be interesting to investigate the relationship with the number theory.4. We analyzed the measure preserving transformations from the dynamical system viewpoint. We studied the Calabi invariant and the differentiability of the dynamical system. In 1996, we found a relationship between the Calabi invariant of the group of arca preserving diffeomorphisms of the disk and the Euler class of the group of diffeomorphisms of the cirele. This should be important in the future investigation. We constructed interesting examples of mcasure prescrving transformations. We continuc to analyze them.5. We asked prof. Hacfliger of the University of Geneva to review our investigation. We got a high mark on our investigation and tion as well as valuable suggestions.