The focused study of automorphism groups of free groups and the mapping class groups for the Johnson homomorphisms

• Project/Area Number: 24740051
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Tokyo University of Science
• Principal Investigator: Satoh Takao 東京理科大学, 理学部, 准教授 (70533256)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 自由群の自己同型群 / Johnson準同型 / 群のコホモロジー / 写像類群 / SL_2 Fricke指標環 / SL_2普遍指標環 / Fricke指標 / IA自己同型群 / 曲面の写像類群 / Torelli群

Outline of Final Research Achievements

We obtained many new results for the Andreadakis-Johnson filtration and the Johnson homomorphisms of the automrophism groups of free groups. In particular, we gave the affirmative answer to the Andreadakis conjecture for the upper triangular automorphism groups, and established an analogue of the Johnson-Morita theory by using the rings of Fricke characters of free groups and the rings of SL(m,C)-representations of free groups. We expect that these results have good applications for the study of the mapping class groups of surfaces.

Research Products

• [Journal Article] On the rings of Fricke characters of free abelian groups (2015)
  • Author(s): Eri Hatakenaka and Takao Satoh
  • Journal Title: Journal of Commutative Algebra Volume: 7 Pages: 545-565
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] The Johnson-Morita theory for the ring of Fricke characters of free groups (2015)
  • Author(s): Takao Satoh
  • Journal Title: Pacific Journal of Mathematics Volume: 275 Pages: 443-461
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] On the graded quotients of the ring of Fricke characters of a free group (2015)
  • Author(s): Eri Hatakenaka and Takao Satoh
  • Journal Title: Journal of Algebra Volume: 430 Pages: 94-118
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] On the basis-conjugating automorphism groups of free groups and free metabelian groups (2015)
  • Author(s): Takao Satoh
  • Journal Title: Mathematical Proceedings Cambridge Philosophical Society Volume: 158 Pages: 83-109
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] On the SL(m,C)-representation rings of free groups and the Johnson homomorphisms (2016)
  • Author(s): Takao Satoh
  • Organizer: Low dimensional topology and number theory VIII
  • Place of Presentation: 九州大学
  • Year and Date: 2016-03-24
  • Int'l Joint Research / Invited
• [Presentation] On the Johnson homomorphism of the basis-conjugating automorphism groups of free groups (2015)
  • Author(s): Takao Satoh
  • Organizer: 表現論による諸問題への新しいアプローチを目指して（第2回）
  • Place of Presentation: 電気通信大学
  • Year and Date: 2015-11-20
  • Invited
• [Presentation] On the Johnson homomorphism of the automorphism groups of free groups (2015)
  • Author(s): Takao Satoh
  • Organizer: Workshop on Automorphism groups of free groups
  • Place of Presentation: University of Copenhagen, Denmark
  • Year and Date: 2015-11-09
  • Int'l Joint Research / Invited
• [Presentation] The Johnson-Morita theory for the rings of Fricke characters (2015)
  • Author(s): Takao Satoh
  • Organizer: Glances@Manifolds
  • Place of Presentation: Uniwersytet Jagiellonski, Poland
  • Year and Date: 2015-07-18
  • Int'l Joint Research / Invited
• [Presentation] On the Andreadakis conjecture of the automorphism groups of free groups (2015)
  • Author(s): Takao Satoh
  • Organizer: RIMS研究集会「新しい変換群論の幾何」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2015-05-28
• [Presentation] On the rings of SL(2,C)-characters of free groups and free abelian groups (2014)
  • Author(s): 佐藤 隆夫
  • Organizer: Towards new applications of representation theory
  • Place of Presentation: 電気通信大学
  • Year and Date: 2014-11-15
  • Invited
• [Presentation] 自由群のFricke指標環とJohnson準同型について (2014)
  • Author(s): 佐藤 隆夫
  • Organizer: 空間の代数的・幾何的モデルとその周辺
  • Place of Presentation: 信州大学
  • Year and Date: 2014-09-18
  • Invited
• [Presentation] 自由群の自己同型群のAndreadakis-Johnson filtrationについて (2014)
  • Author(s): 佐藤 隆夫
  • Organizer: 空間の代数的・幾何的モデルとその周辺
  • Place of Presentation: 信州大学
  • Year and Date: 2014-09-17
  • Invited
• [Presentation] Some problems on the Johnson homomorphisms and related topics (2014)
  • Author(s): Takao Satoh
  • Organizer: Workshop on Algebraic Topology and Its Applications
  • Place of Presentation: 大連理工大学
  • Year and Date: 2014-08-07
  • Invited
• [Presentation] On the Johnson homomorphisms of automorphism groups of free groups (2014)
  • Author(s): Takao Satoh
  • Organizer: Workshop on Algebraic Topology and Its Applications
  • Place of Presentation: 大連理工大学
  • Year and Date: 2014-08-06
  • Invited
• [Presentation] A representation theoretic approach to the Johnson cokernels I
  • Author(s): 佐藤 隆夫
  • Organizer: Workshop: Johnson homomorphisms
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Invited
• [Presentation] On the rings of SL(2,C)-characters of free groups
  • Author(s): 佐藤 隆夫
  • Organizer: Geometry of Manifolds and Group Actions
  • Place of Presentation: Adam Mickiewicz University, Slubice, Poland
• [Presentation] On the rings of SL(2,C)-characters of free groups
  • Author(s): 佐藤 隆夫
  • Organizer: East Asian Conference on Algebraic Topology 5
  • Place of Presentation: 中国科学院数学与系統科学研究院
  • Invited
• [Presentation] On the rings of SL(2,C)-characters of free abelian groups
  • Author(s): 佐藤 隆夫
  • Organizer: 第40回変換群論シンポジウム
  • Place of Presentation: 明治大学中野キャンパス
• [Presentation] The Johnson-Morita theory for the rings of Fricke characters of free groups
  • Author(s): 佐藤 隆夫
  • Organizer: New Trends in Teichmuller Theory and Mapping Class Groups
  • Place of Presentation: オーベルヴォルファッハ数学研究所，ドイツ
  • Invited
• [Presentation] 自由群の自己同型群とFricke指標環について
  • Author(s): 佐藤 隆夫
  • Organizer: RIMS研究集会「変換群の幾何の展開」
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] 自由群の自己同型群のJohnson準同型写像について
  • Author(s): 佐藤 隆夫
  • Organizer: 代数・幾何セミナー
  • Place of Presentation: 慶應義塾大学理工学部数理科学科
  • Invited
• [Presentation] On the Johnson cokernels of the mapping class group of a surface
  • Author(s): Satoh, Takao
  • Organizer: Group actions and applications in geometry, topology and analysis
  • Place of Presentation: Kunming University of Science and Technology, China
• [Presentation] 自由群の自己同型群とFricke指標環について
  • Author(s): 佐藤 隆夫
  • Organizer: 数理科学セミナー
  • Place of Presentation: 東京都市大学
• [Presentation] On the augmentation quotients of the IA-automorphism group of a free group
  • Author(s): 佐藤 隆夫
  • Organizer: 談話会
  • Place of Presentation: 京都大学大学院理学研究科数学教室
• [Presentation] On the automorphism groups of free groups and the ring of Fricke characters
  • Author(s): 佐藤 隆夫
  • Organizer: Mini workshop in Singularities
  • Place of Presentation: 東京理科大学理学部数学科