A research of the automorphism group of a free group

• Project/Area Number: 21740050
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: Tokyo University of Science (2011); Kyoto University (2009-2010)
• Principal Investigator: SATOH Takao 東京理科大学, 理学部第二部・数学科, 講師 (70533256)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2009: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 自由群の自己同型群 / Johnson準同型 / 曲面の写像類群 / IA-自己同型群 / Magnus表現

Research Abstract

Our main results of this research are as follows : (1) We determined the cokernel of the Johnson homomorphism associated to the lower central series of the IA-automorphism group of a free group. (2) We gave combinatorial descriptions of the GL-decomposition of the Johnson cokernels and the derivation algebra of the free Lie algebra. (3) We constructed new series of obstructions for the surjectivity of the Johnson homomorphism of the mapping class group of a surface. (4) We computed the first cohomology group of the automorphism group of a free group with coefficients in the abelianization of the IA-automorphism group.

Research Products

• [Journal Article] On the Johnson filtration of the basis-conjugating automorphism group of a free group (2012)
  • Author(s): Takao Satoh
  • Journal Title: Michigan Mathematical Journal Volume: 61 Pages: 87-105
  • Peer Reviewed
• [Journal Article] On the lower central series of the IA-automorphism group of a free group (2012)
  • Author(s): Takao Satoh
  • Journal Title: Journal of Pure and Applied Algebra Volume: 216 Pages: 709-717
  • Peer Reviewed
• [Journal Article] A reduction of the target of the Johnson homomorphisms of the automorphism group of a free group (2011)
  • Author(s): Takao Satoh
  • Journal Title: Transactions of American Mathematical Society Volume: 363 Pages: 1631-1664
  • NAID: 120002970847
  • Peer Reviewed
• [Journal Article] The Johnson filtration of the McCool stabilizer subgroup of the automorphism group of a free group (2011)
  • Author(s): Takao Satoh
  • Journal Title: Proc. of Amer. Math. Soc Volume: 139 Pages: 1237-1245
  • Peer Reviewed
• [Journal Article] On the lower central quotients of the free groups of finite rank of a certain variety (2011)
  • Author(s): Takao Satoh
  • Journal Title: Journal of Algebra Volume: 331 Pages: 449-465
  • Peer Reviewed
• [Journal Article] The kernel of the Magnus representation of the automorphism group of a free group is not finitely generated (2011)
  • Author(s): Takao Satoh
  • Journal Title: Mathematical Proceedings Cambridge Philosophical Society Volume: 151 Pages: 407-419
  • Peer Reviewed
• [Journal Article] On the derivation algebra of the free Lie algebra and trace maps (2011)
  • Author(s): Naoya Enomoto and Takao Satoh
  • Journal Title: Algebraic and Geometric Topology Volume: 11 Pages: 2861-2901
  • Peer Reviewed
• [Journal Article] The Johnson filtration of the McCool stabilizer subgroup of the automorphism group of a free group (2011)
  • Author(s): Takao Satoh
  • Journal Title: Proc.of Amer.Math.Soc. Volume: 139 Pages: 1237-1245
  • Peer Reviewed
• [Journal Article] On the image of the Burau representation of the IA-automorphism group of a free group (2010)
  • Author(s): Takao Satoh
  • Journal Title: Journal of Pure and Applied Algebra Volume: 214 Pages: 584-595
  • NAID: 120001823760
  • Peer Reviewed
• [Journal Article] On the fourth Johnson homomorphism of the automorphism group of a free group (2010)
  • Author(s): Takao Satoh
  • Journal Title: Journal of Algebra Volume: 323 Pages: 3182-3201
  • Peer Reviewed
• [Journal Article] On the image of the Burau representation of the IA-automorphism group of a free group (2010)
  • Author(s): Takao Satoh
  • Journal Title: Journal of Pure and Applied Algebra 214 Pages: 584-594
  • NAID: 120001823760
  • Peer Reviewed
• [Presentation] n the augmentation quotients of the IA-automorphism group of a free grou (2012)
  • Author(s): 佐藤隆夫
  • Organizer: トポロジーセミナー
  • Place of Presentation: 東京工業大学
  • Year and Date: 2012-12-05
• [Presentation] On the augmentation quotients of the IA-automorphism group of a free group (2012)
  • Place of Presentation: 東京工業大学トポロジーセミナー
  • Year and Date: 2012-01-25
• [Presentation] On the Johnson cokernels of the mapping class group of a surface (2012)
  • Organizer: トポロジー火曜セミナー
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Year and Date: 2012-01-17
• [Presentation] On the derivation algebra of the free Lie algebra, and trace maps (2011)
  • Organizer: 第38回変換群論シンポジウム
  • Place of Presentation: 兵庫県民会館
  • Year and Date: 2011-11-20
• [Presentation] 曲面の写像類群のJohnson余核について (2011)
  • Organizer: 城崎ホモトピー論研究会
  • Place of Presentation: 城崎町健康福祉センター
  • Year and Date: 2011-11-04
• [Presentation] 自由リー代数の微分代数とTrace写像について (2011)
  • Organizer: 幾何学コロキウム
  • Place of Presentation: 北海道大学理学部数学科
  • Year and Date: 2011-09-12
• [Presentation] 自由群の自己同型群のねじれ係数コホモロジー群について (2011)
  • Organizer: RIMS研究集会「有限群のコホモロジー論とその周辺」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2011-09-01
• [Presentation] 曲面の写像類群のJohnson準同型の余核について (2011)
  • Organizer: RIMS研究集会「群の表現及び表現空間の幾何学的・解析的研究」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2011-06-23
• [Presentation] 曲面の写像類群のJohnson準同型の余核について (2011)
  • Organizer: RIMS研究集会「変換群の幾何と組合せ論」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2011-06-16
• [Presentation] 曲面の写像類群のJohnson準同型の余核について (2011)
  • Organizer: 特異点セミナー
  • Place of Presentation: 東京理科大学理学部数学科
  • Year and Date: 2011-05-21
• [Presentation] On the cokernel of the Johnson homomorphism of the mapping class group of a surface (2011)
  • Organizer: Branched Coverings, Degenerations, and Related Topics
  • Place of Presentation: 首都大学東京
  • Year and Date: 2011-03-08
• [Presentation] On the cokernel of the Johnson homomorphism of the mapping class group of a surface (2011)
  • Author(s): Takao Satoh
  • Organizer: Branched Coverings, Degenerations, and Related Topics
  • Place of Presentation: 首都大学東京(招待講演)
  • Year and Date: 2011-03-08
• [Presentation] 自由群の自己同型群のねじれ係数コホモロジーについて (2011)
  • Organizer: 北田先生を囲む研究集会
  • Place of Presentation: 京都市大学のまち交流センター
  • Year and Date: 2011-02-12
• [Presentation] 自由群の自己同型群のねじれ係数コホモロジーについて (2011)
  • Author(s): 佐藤隆夫
  • Organizer: 北田先生を囲む研究集会
  • Place of Presentation: 京都市大学のまち交流センター(招待講演)
  • Year and Date: 2011-02-12
• [Presentation] 自由群の自己同型群のねじれ係数コホモロジーについてII (2011)
  • Place of Presentation: 信州大学理学部数理・自然情報科学教室
  • Year and Date: 2011-01-12
• [Presentation] 自由群の自己同型群のねじれ係数コホモロジーについてII (2011)
  • Author(s): 佐藤隆夫
  • Organizer: 談話会
  • Place of Presentation: 信州大学理学部数理・自然情報科学教室(招待講演)
  • Year and Date: 2011-01-12
• [Presentation] 自由群の自己同型群のねじれ係数コホモロジーについてI (2011)
  • Organizer: 信州大学理学部数理・自然情報科学教室
  • Place of Presentation: トポロジーセミナー
  • Year and Date: 2011-01-11
• [Presentation] 自由群の自己同型群のねじれ係数コホモロジーについてI (2011)
  • Author(s): 佐藤隆夫
  • Organizer: トポロジーセミナー
  • Place of Presentation: 信州大学理学部数理・自然情報科学教室(招待講演)
  • Year and Date: 2011-01-11
• [Presentation] Teichmuller Theory (2010)
  • Organizer: On the Johnson homomorphisms of the automorphism group of a free group
  • Place of Presentation: Oberwolfach
  • Year and Date: 2010-12-04
• [Presentation] On the Johnson homomorphisms of the automorphism group of a free group (2010)
  • Author(s): Takao Satoh
  • Organizer: Teichmuller Theory
  • Place of Presentation: Oberwolfach, German(招待講演)
  • Year and Date: 2010-12-04
• [Presentation] 自由群の自己同型群のJohnson filtrationについて (2010)
  • Organizer: 日本数学会秋季総合分科会
  • Year and Date: 2010-09-22
• [Presentation] 自由群の自己同型群のJohnson filtrationについて (2010)
  • Author(s): 佐藤隆夫
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-22
• [Presentation] On the kernel of the Magnus representation of the automorphism group of a free group (2010)
  • Organizer: 研究集会「リーマン面に関連する位相幾何学」
  • Place of Presentation: 東大数理
  • Year and Date: 2010-09-06
• [Presentation] On the kernel of the Magnus representation of the automorphism group of a free group (2010)
  • Author(s): 佐藤隆夫
  • Organizer: 研究集会「リーマン面に関連する位相幾何学」
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Year and Date: 2010-09-06
• [Presentation] On the Johnson filtration of the automorphism group of a free group (2010)
  • Organizer: RIMS研究集会「変換群と手術理論」
  • Place of Presentation: 京大数理研
  • Year and Date: 2010-09-01
• [Presentation] On the Johnson filtration of the automorphism group of a free group (2010)
  • Author(s): 佐藤隆夫
  • Organizer: 研究集会「変換群と手術理論」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-09-01
• [Presentation] 自由群の自己同型群のMagnus表現の核について (2010)
  • Organizer: 日本数学会春の年会
  • Place of Presentation: 慶應義塾大学矢上キャンパス
  • Year and Date: 2010-03-24
• [Presentation] Geometry and Analysis (2009)
  • Organizer: On the Johnson homomorphisms of the automorphism group of a free group
  • Place of Presentation: Paris
  • Year and Date: 2009-12-21
• [Presentation] On the Johnson homomorphisms of the automorphism group of a free group (2009)
  • Author(s): Takao Satoh
  • Organizer: Geometry and Analysis
  • Place of Presentation: パリ第7大学(フランス)
  • Year and Date: 2009-12-21
• [Presentation] On the kernel of the Magnus representation of the automorphism group of a free group (2009)
  • Organizer: 研究集会「第36回変換群論シンポジウム」
  • Place of Presentation: 大阪市立大学学術情報総合センター
  • Year and Date: 2009-12-10
• [Presentation] On the kernel of the Magnus representation of the automorphism group of a free group (2009)
  • Author(s): 佐藤隆夫
  • Organizer: 第36回変換群論シンポジウム
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2009-12-10
• [Presentation] On the Johnson filtration of the automorphism group of a free group (2009)
  • Organizer: 研究集会「微分・代数トポロジーの現在と未来」
  • Place of Presentation: かんぽの宿徳島
  • Year and Date: 2009-11-11
• [Presentation] On the Johnson filtration of the automorphism group of a free group (2009)
  • Author(s): 佐藤隆夫
  • Organizer: 微分・代数トポロジーの現在と未来
  • Place of Presentation: かんぽの宿 徳島
  • Year and Date: 2009-11-11
• [Presentation] 自由群の自己同型群のねじれ係数コホモロジー群について (2009)
  • Organizer: 日本数学会秋の年会
  • Place of Presentation: 大阪大学豊中キャンパス
  • Year and Date: 2009-09-26
• [Presentation] 自由群の自己同型群のねじれ係数コホモロジー群について (2009)
  • Author(s): 佐藤隆夫
  • Organizer: 日本数学会秋の年会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-26
• [Presentation] 自由群の自己同型群のJohnson準同型について (2009)
  • Organizer: 研究集会「群と力学系に関わる離散幾何学」
  • Place of Presentation: 愛媛大学理学部数学科
  • Year and Date: 2009-09-02
• [Presentation] 自由群の自己同型群のJohnson準同型について (2009)
  • Author(s): 佐藤隆夫
  • Organizer: 群と力学系に関わる離散幾何学
  • Place of Presentation: 愛媛大学理学部数学科
  • Year and Date: 2009-09-02
• [Presentation] 自由群の自己同型群のJohnson準同型について (2009)
  • Organizer: 京都大学大学院理学研究科数学教室談話会
  • Place of Presentation: 京都大学大学院理学研究科数学教室
  • Year and Date: 2009-05-27
• [Presentation] 自由群の自己同型群のJohnson準同型について (2009)
  • Author(s): 佐藤隆夫
  • Organizer: 京都大学大学院理学研究科数学教室談話会
  • Place of Presentation: 京都大学大学院理学研究科
  • Year and Date: 2009-05-27
• [Remarks]
  • URL: http://www.math.kyoto-u.ac.jp/~takao/