On the low dimensional cohomology groups of the IA-automorphism group of a free group of rank three

2016/07/19 by Takao Satoh, Satoh, Takao
Mathematics · #20F28 #Advanced Algebra and Geometry #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1607.05393

openalex publication_date 2016/07/19 · openalex created_date 2016/08/23 · openalex updated_date 2026/07/28

Abstract

In this paper we study the structure of the rational cohomology groups of the IA-automorphism group IA₃ of a free group of rank three by using combinatorial group theory and representation theory. In particular, we detect non-trivial irreducible component in the second cohomology group of IA₃, which does not contained in the image of the cup product map of the first cohomology groups. We also show that the image of the triple cup product map of the first cohomology groups in the third cohomology group is trivial. As a corollary, we obtain that the fourth term of the lower central series of IA₃ has finite index in that of the Andreadakis-Johnson filtration.

Related

• Quotient groups of IA-automorphisms of a free group of rank 3
• On the Andreadakis-Johnson filtration of the automorphism group of a free group
• First cohomology groups of the automorphism group of a free group with coefficients in the abelianization of the IA-automorphism group
• The walled Brauer category and stable cohomology of IA_n
• On the stable cohomology of the IA-automorphism groups of free groups