Rigidity of higher dimensional Lie groups -- Toward comprehensive understanding

• Project/Area Number: 18K03294
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: The University of Tokyo
• Principal Investigator: Kanai Masahiko 東京大学, 大学院数理科学研究科, 教授 (70183035)
• Project Period (FY): 2018-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2020)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2019: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2018: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: 剛性 / 高階リー群 / トンプソン群 / 高階リー群の剛性 / Margulis の超剛性定理 / Mostow の超剛性定理 / 接調和写像 / 実階数が2 以上の非コンパクト半単純リー群 / 松島の消滅定理

Outline of Final Research Achievements

It is known that semi-simple Lie groups of real-rank greater than or equal to 2 exhibit rigidity phenomena. The proofs of such results are often done by appealing to a classification of such Lie groups (or their Lie algebras). The present research aims at unifying those theorems. I gave a new proof of Matsushima's vanishing based on such a scenario. A basic strategy is -- For a certain foliated space, apply the theory of harmonic integration in the direction tangent to the foliation, and ergodic theory in the transverse direction.
We made investigations on the Thompson group F, as well. In particular, I discovered a deep similarity between a result on the automorphism group of F done by Brin et al., and a new infinite-dimensial space on which F acts.

Academic Significance and Societal Importance of the Research Achievements

最終年度にコロナ禍に襲われたこともあり，本研究計画は完成にはほど遠いと言わざるをえない．しかし，すでに部分的な結果は得られている．それらを発展させ，さらに新たな考察を積み上げれば，この分野の専門家たちから十分に高い評価を得られるのではないかと期待している．

Research Products

• [Funded Workshop] Rigidity School, 2018. (2018)