Geometric study on infinite simple groups

• Project/Area Number: 21654009
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: The University of Tokyo
• Principal Investigator: TSUBOI Takashi 東京大学, 大学院・数理科学研究科, 教授 (40114566)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥3,300,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥300,000); Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2010: ¥1,000,000 (Direct Cost: ¥1,000,000); Fiscal Year 2009: ¥1,000,000 (Direct Cost: ¥1,000,000)
• Keywords: 無限単純群 / 微分同相群 / 完全群 / 幾何学的群論 / 幾何学 / トポロジー / 代数学

Research Abstract

We studied infinite simple groups as geometric objects where they have verious quasimorphisms and the distance function d on the set of unions of a conjugate class and its inverse. We determined the quasi isometry type with respect to the distance d of the infinite alternative group A_infty. We showed and published the result that the identity component of the group of real analytic diffeomorphisms of a manifold with good circle actions is perfect. We showed and published the result that the identity component of the group of diffeomorphisms of a closed connected manifold of dimensions not eaual to 2 and 4 is uniformly perfect and uniformly simple. We also studied the homeomorphism groups of commutator width one.

Research Products

• [Journal Article] On the uniform perfectness of the groups of diffeomorphisms ofeven-dimensional manifolds (2012)
  • Author(s): Takashi Tsuboi
  • Journal Title: CommentariiMathematici Helvetici Volume: 87 Issue: 1 Pages: 141-185
  • DOI: 10.4171/cmh/251
  • Peer Reviewed
• [Journal Article] On the uniform simplicity of diffeomorphism groups, Differential Geometry, Proceedings of the VIII International Colloquium, Santiago de Compostela, 2008 (2009)
  • Author(s): Takashi Tsuboi
  • Journal Title: World Scientific, Singapore Pages: 43-45
  • Peer Reviewed
• [Journal Article] On the group of real analytic diffeomorphisms (2009)
  • Author(s): Takashi Tsuboi
  • Journal Title: Annales Scientifiques de l' Ecole Normale Superieure Volume: 49 Pages: 601-651
  • Peer Reviewed
• [Journal Article] Classifying spaces for groupoid structures, Foliations, Geometry, and Topology : Paul Schweitzer Festschrift (2009)
  • Author(s): Takashi Tsuboi
  • Journal Title: Contemporary Mathematics Volume: 498 Pages: 67-81
  • Peer Reviewed
• [Journal Article] Classifying spaces for groupoid structures (2009)
  • Author(s): Takashi Tsuboi
  • Journal Title: Contemporary Mathematics 498 Pages: 67-81
  • Peer Reviewed
• [Journal Article] On the group of real analytic diffeomorphisms (2009)
  • Author(s): Takashi Tsuboi
  • Journal Title: Annales Scientifiques de l'Ecole Normale Superieure 49 Pages: 601-651
  • Peer Reviewed
• [Journal Article] On the uniform simplicity of diffeomorphism groups (2009)
  • Author(s): Takashi Tsuboi
  • Journal Title: Proceedings of the VIII International Colloquium Pages: 43-55
  • Peer Reviewed
• [Presentation] Homeomorphism groups of commutator width one, Poster at Geometry and Dynamics, Todai Forum (2011)
  • Author(s): Takashi TSUBOI
  • Organizer: Ecole Normale Superieure de Lyon
  • Place of Presentation: France
  • Year and Date: 2011-10-17
• [Presentation] Homeomorphism groups of commutator width one (2011)
  • Author(s): Takashi Tsuboi
  • Organizer: Geometry and Dynamics, Todai Forum
  • Place of Presentation: Ecole Normale Superieure de Lyon (France)
  • Year and Date: 2011-10-17
• [Presentation] On the uniform perfectness of the group of diffeomorphisms (2010)
  • Author(s): akashi TSUBOI
  • Organizer: Centro de Recerca Matematica
  • Place of Presentation: Barcelona, Spain
  • Year and Date: 2010-07-13
• [Presentation] On the uniform perfectness of the group of diffeomorphisms (2010)
  • Author(s): Takashi Tsuboi
  • Organizer: Conference on Geometry and Topology of Foliations
  • Place of Presentation: CRM, Barcelona,スペイン
  • Year and Date: 2010-07-13
• [Presentation] Diff^ω(CP^2)_0,「複素解析的ベクトル場・葉層構造とその周辺」 (2009)
  • Author(s): 坪井俊
  • Organizer: 研究集会
  • Place of Presentation: 龍谷大学セミナーハウスともいき荘
  • Year and Date: 2009-12-13
• [Presentation] ${Diff}^\omega(CP^2)_0$ (2009)
  • Author(s): 坪井俊
  • Organizer: 「複素解析的ベクトル場・葉層構造とその周辺」研究集会
  • Place of Presentation: 龍谷大学セミナーハウスともいき荘
  • Year and Date: 2009-12-13