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

TL;DR

This paper investigates the rational cohomology groups of the IA-automorphism group of a rank three free group, revealing new irreducible components and properties of cup product maps, with implications for the group's lower central series.

Contribution

It identifies a non-trivial irreducible component in the second cohomology of IA_3 and shows the triviality of the triple cup product map in the third cohomology, advancing understanding of IA_3's structure.

Findings

Detected a non-trivial irreducible component in H^2 of IA_3.

Proved the triple cup product map in H^3 is trivial.

Established that the fourth term of the lower central series has finite index in the Andreadakis-Johnson filtration.

Abstract

In this paper we study the structure of the rational cohomology groups of the IA-automorphism group $\mathrm{IA}_3$ 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 $\mathrm{IA}_3$, 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 $\mathrm{IA}_3$ has finite index in that of the Andreadakis-Johnson filtration.