Study on cohomologies of automorphism groups of free groups focued on the unstable range

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03477
• 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: Tokyo University of Science
• Principal Investigator: Satoh Takao 東京理科大学, 理学部第二部数学科, 教授 (70533256)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 自由群の自己同型群 / 群のコホモロジー / ジョンソン準同型

Outline of Final Research Achievements

In this research, we studied twisted cohomology groups of the automorphism groups of free groups on the unstable range. We computed all the cohomology groups with coefficients in the exterior products of the abelianization of the free group in the case where the rank of the free group is three. In particular, we verified that a certain degeneration is happen in some second cohomology groups, compare to those in the stable range. We also studied the first cohomology groups with coefficients in the abelianization of a certain verbal subgroup of the free group. From our previous works, it is easily seen that there are two linearly independent cohomology classes constructed from the Morita cocycles. By our computation, we verified that in the case where the rank of the free group is two, the rank of the first cohomology group is more than two in general.

Free Research Field

代数的位相幾何学

Academic Significance and Societal Importance of the Research Achievements

この半世紀の間に自由群の自己同型群のホモロジーに関する研究は急速に進展を遂げており，安定域に関しては自由群のアーベル化を経由するようなねじれ係数ホモロジーの計算方法が確立されている．一方で，非安定域のねじれ係数ホモロジー群の構造については体系的な研究手法は未だ確立しておらず，具体的な計算結果を蓄積してその振る舞いを調べていくしかないのが現状である．本研究において，自由群の階数が3の場合の計算結果や，自由群のアーベル化を経由しないような係数のホモロジー群についていくつかの新しい結果が得られたことは一定の学術的意義があると考えている．