STUDY ON INFINITE TRANSFORMATION GROUPS ON MANIFOLDS

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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.