2022 Fiscal Year Final Research Report
Geometry of discrete groups and its applications to 3-dimensional topology
| Project/Area Number |
17H02843
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Gakushuin University (2018-2022)
Osaka University (2017) |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
作間 誠 大阪公立大学, 数学研究所, 特別研究員 (30178602)
宮地 秀樹 金沢大学, 数物科学系, 教授 (40385480)
山下 靖 奈良女子大学, 自然科学系, 教授 (70239987)
森藤 孝之 慶應義塾大学, 経済学部(日吉), 教授 (90334466)
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| Project Period (FY) |
2017-04-01 – 2022-03-31
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| Keywords | ３次元多様体 / Klein群 / Teichmuller空間 / 離散群 |
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| Outline of Final Research Achievements |
We studied Kleinian groups and Teichmuller spaces from geometric viewpoint aiming at applications to three-dimensional topology. In the field of Kleinian group theory, we studied continuity/discontinuity of Cannon-Thurston maps in the deformation spaces of Kleinian surface groups, and gave complete criteria for the discontinuity of the maps. We also showed that a given data of bending laminations and ending laminations with reasonable necessary conditions can always be realised by a Kleinian surface group.
In the field of Teichmuller theory, we studied Thurston's asymmetric metric on Teichmuller space, and showed in particular its infinitesimal rigidity. We also considered the case of torus, and showed that there is a natural continuous deformation from the Teichmuller metric to Thurston's asymmetric metric.
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| Free Research Field |
位相幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
３次元トポロジーは現代の幾何学において主要な研究対象であるが，その理解と発展のためには，離散群，とりわけKlein群とTeichmuller空間の理論を発展させることが不可欠である．本研究はそのような動機で，Klein群とTeichmuller空間の研究を続けた．今回の研究では1980年代のThurstonの研究以来の懸案の問題をいくつか解決することに成功している．
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