On the low dimensional cohomology groups of the IA-automorphism group of the free group of rank three (Cohomology theory of finite groups and related topics)

Abstract

In this announcement we consider the structure of the rational cohomology groups of the IA-automorphism group mathrm{I}mathrm{A}{3} of the free group of rank three by using combinatorial group theory and representation theory. In particular, we detect a non-trivial irreducible component in the second cohomology group of mathrm{I}mathrm{A}{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{I}mathrm{A}{3} has finite index in that of the Andreadakis-Johnson filtration.