STUDY ON INFINITE TRANSFORMATION GROUPS ON MANIFOLDS
Project/Area Number  07454013 
Research Category 
GrantinAid for Scientific Research (B)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  THE UNIVERSITY OF TOKYO 
Principal Investigator 
TSUBOI Takashi Univ. of Tokyo, Graduate School of Math. Sci. Prof., 大学院・数理科学研究科, 教授 (40114566)

CoInvestigator(Kenkyūbuntansha) 
MATANO Hiroshi Univ. of Tokyo, Graduate School of Math. Sci. Prof., 大学院・数理科学研究科, 教授 (40126165)
ODA Takayuki Univ. of Tokyo, Graduate School of Math. Sci. Prof., 大学院・数理科学研究科, 教授 (10109415)
OSHIMA Toshio Univ. of Tokyo, Graduate School of Math. Sci. Prof., 大学院・数理科学研究科, 教授 (50011721)
MATSUMOTO Yukio Univ. of Tokyo, Graduate School of Math. Sci. Prof., 大学院・数理科学研究科, 教授 (20011637)
OCHIAI Takushiro Univ. of Tokyo, Graduate School of Math. Sci. Prof., 大学院・数理科学研究科, 教授 (90028241)
斎藤 毅 東京大学, 大学院・数理科学研究科, 助教授 (70201506)
川又 雄二郎 東京大学, 大学院・数理科学研究科, 教授 (90126037)

Project Period (FY) 
1995 – 1996

Project Status 
Completed(Fiscal Year 1996)

Budget Amount *help 
¥5,500,000 (Direct Cost : ¥5,500,000)
Fiscal Year 1996 : ¥2,600,000 (Direct Cost : ¥2,600,000)
Fiscal Year 1995 : ¥2,900,000 (Direct Cost : ¥2,900,000)

Keywords  finitely presented groups / diffeomorphism groups / limit sets / dynamical systems / infinite transformation groups / manifolds / the Calabi invariant / 保測変換 
Research Abstract 
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.

Report
(3results)
Research Products
(15results)