Global geometric studies on Kleinian groups and Teichmuller spaces

2015 Fiscal Year Final Research Report

• Project/Area Number: 22244005
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Osaka University
• Principal Investigator: Ohshika Ken'ichi 大阪大学, 理学(系)研究科(研究院), 教授 (70183225)
• Co-Investigator(Kenkyū-buntansha): Miyachi Hideki 大阪大学, 大学院理学研究科, 准教授 (40385480) ; Soma Teruhiko 首都大学, 東京大学院理工学研究科, 教授 (50154688) ; Wada Masaaki 大阪大学, 大学院情報科学研究科, 教授 (80192821) ; Endo Hisaaki 東京工業大学, 大学院理工学研究科, 教授 (20323777) ; Kawazumi Nariya 東京大学, 大学院数理科学研究科, 准教授 (30214646)
• Project Period (FY): 2010-04-01 – 2016-03-31
• Keywords: Klein群 / Teichmuller空間 / コンパクト化 / 変形空間

Outline of Final Research Achievements

We first worked on constructing fundamental theory of Kleinian groups, which centred around 10 problems raised by Thurston in 1980's. In that process, we succeeded in resolving two among these problems completely. Based upon this work, we studied deformation spaces of Kleinian groups from geometric viewpoint. We could get new insights into the topology of the boundaries of deformation spaces, and at the same time, succeeded in getting a necessary and sufficient condition on the primitive stability, studying dynamical properties of deformation spaces embedded in character varieties. Applying what we got in the theory of Kleinian groups to the Teichmuller theory, we studied reduced compactification of Teichmuller spaces, which should serve as a common ground for various compactifications of Teichmuller spaces.

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