Studies of higher twisted cohomology groups of automorphism groups of free groups

• Project/Area Number: 16K05155
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Tokyo University of Science
• Principal Investigator: Satoh Takao 東京理科大学, 理学部第二部数学科, 准教授 (70533256)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 自由群の自己同型群 / 群のコホモロジー / 非安定コホモロジー / Andreadakis予想 / Outer space / 群のホモロジー / Johnson準同型 / 群の表現論 / 代数的位相幾何学

Outline of Final Research Achievements

We obtained three main results. The first one is that we found a new GL-irreducible component in the second homology group of the IA-automorphism group of the free group of rank 3. The second one is that we gave the affirmative answer to the Andreadakis problem in the degree three case. The third one is that we obtained non-trivial unstable twisted second cohomology classes of automorphism groups of free groups by using Kawazumi's 1-cocycles.

Academic Significance and Societal Importance of the Research Achievements

近年，自由群の自己同型群のコホモロジーの研究に関しては，非安定域に大きな研究の関心が集まっており，階数3の場合のIA自己同型群のホモロジーの新しいGL既約表現や，自由群の自己同型群の非安定2次元コホモロジー類の構成などは，最近の最先端の研究情勢から鑑みても相応のインパクトがあるものと確信している．
また，Andreadakis予想は50年以上も未解決問題であり，一部分とはいえその解決に寄与できたことは，一般の場合の解決に向けた一つの足掛かりを与えたことでもあり，大変有意義であると考えている．

Research Products

• [Journal Article] The third subgroup of the Andreadakis-Johnson filtration of the automorphism group of a free group (2019)
  • Author(s): Takao Satoh
  • Journal Title: Journal of Group Theory Volume: 22 Pages: 41-61
  • Peer Reviewed
• [Journal Article] The third subgroup of the Andreadakis-Johnson filtration of the automorphism group of a free group (2019)
  • Author(s): Takao Satoh
  • Journal Title: J. Group Theory Volume: 22 Pages: 41-61
  • Peer Reviewed
• [Presentation] On the Andreadakis conjecture of the automorphism groups of free groups (2019)
  • Author(s): Takao Satoh
  • Organizer: Johnson homomorphisms and related topics 2019
  • Int'l Joint Research / Invited
• [Presentation] On the Andreadakis conjecture of the automorphism groups of free groups (2019)
  • Author(s): Takao Satoh
  • Organizer: 京都大学大学院理学研究科数学教室代数トポロジーセミナー
  • Invited
• [Presentation] On twisted cohomology groups of the automorphism groups of free groups (2018)
  • Author(s): Takao Satoh
  • Organizer: Joint meeting of the Italian Mathematical Union, the Italian Society of Industrial and Applied Mathematics, the Polish Mathematical Society
  • Int'l Joint Research / Invited
• [Presentation] On the Andreadakis conjecture of the automorphism groups of free groups (2018)
  • Author(s): Takao Satoh
  • Organizer: 京都大学数理解析研究所研究集会・変換群論における幾何・代数・組み合わせ論
  • Invited
• [Presentation] On some representations of the automorphism groups of free groups (2018)
  • Author(s): Takao Satoh
  • Organizer: Workshop on Geometric Discrete Mathematics II
  • Int'l Joint Research / Invited
• [Presentation] On the Andreadakis conjecture of the automorphism groups of free groups (2018)
  • Author(s): Takao Satoh
  • Organizer: Topology of Arrangements and Representation stability, Mathematisches Forschungsinstitut Oberwolfach
  • Int'l Joint Research / Invited
• [Presentation] On the Andreadakis conjecture of the automorphism groups of free groups (2018)
  • Author(s): Takao Satoh
  • Organizer: Representation spaces, Teichmüller theory, and their relationship with 3-manifolds from the classical and quantum viewpoints, Universite d'Aix-Marseille
  • Int'l Joint Research / Invited
• [Presentation] On the Andreadakis conjecture of the automorphism groups of free groups (2018)
  • Author(s): Takao Satoh
  • Organizer: Colloquium, The University of Gottingen
  • Int'l Joint Research
• [Presentation] On the cohomology groups of the IA-automorphism groups of free groups of rank three (2017)
  • Author(s): 佐藤 隆夫
  • Organizer: 有限群のコホモロジーとその周辺
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2017-02-16
• [Presentation] On the Andreadakis conjecture of the automorphism groups of free groups (2017)
  • Author(s): Takao Satoh
  • Organizer: Colloquium talk at the Faculty of Mathematics and Computer Sciences, Adam Mickiewicz University
  • Invited
• [Presentation] On the Andreadakis conjecture of the automorphism groups of free groups (2017)
  • Author(s): Takao Satoh
  • Organizer: Algebraic Topology seminar, The University of Strasbourg
  • Invited
• [Presentation] On the cohomology groups of the IA-automorphism groups of free groups of rank three (2016)
  • Author(s): 佐藤 隆夫
  • Organizer: 群作用と位相
  • Place of Presentation: 城崎健康福祉センター（兵庫県豊岡市城崎町）
  • Year and Date: 2016-12-06
• [Presentation] On the cohomology groups of the IA-automorphism groups of free groups of rank three (2016)
  • Author(s): 佐藤 隆夫
  • Organizer: 変換群論シンポジウム2016
  • Place of Presentation: 姫路市市民会館（兵庫県姫路市）
  • Year and Date: 2016-11-17
• [Presentation] On the cohomology groups of the IA-automorphism groups of free groups of rank three (2016)
  • Author(s): 佐藤 隆夫
  • 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3回）
  • Place of Presentation: 電気通信大学
  • Year and Date: 2016-11-11