STUDY ON INFINITE TRANSFORMATION GROUPS ON MANIFOLDS

1. 我们研究了群有限生成和有限呈现的条件。我们给出了群作用于单连通空间的证明，使得商空间是有限单形，各向同性群是有限生成和有限呈现的。我们还收集了很多关于有限生成和有限呈现组的知识。对微分同态群的分类空间的研究扩展到对极限集或端点集的同胚群的分类空间的研究。我们得到了关于它们拓扑结构的几个结果。特别地，我们证明了Menger紧空间的同胚群的分类空间是无环的。我们于1995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。我们看了保形变换群的极限集。我们构造了几个离散共形变换群的例子。研究它与数论的关系将是很有趣的。从动力系统的角度分析了保持变换的测度。我们研究了卡拉比不变量和动力系统的可微性。1996年，我们发现了圆盘的保弧微同构群的Calabi不变量与圆的微同构群的欧拉类之间的关系。这在今后的研究中是很重要的。我们构造了一些有趣的保持度量变换的例子。我们继续分析它们。我们请日内瓦大学的Hacfliger教授审查我们的调查。我们的调查和工作得到了很高的评价，并提出了宝贵的建议。

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.