On the Andreadakis conjecture restricted to the “lower-triangular” automorphism groups of free groups

Abstract

In this paper, we consider a certain subgroup [Formula: see text] of the IA-automorphism group of a free group. We determine the images of the [Formula: see text]th Johnson homomorphism restricted to [Formula: see text] for any [Formula: see text] and [Formula: see text]. By using this result, we give an affirmative answer to the Andreadakis conjecture restricted for [Formula: see text]. Namely, we show that the intersection of the Andreadakis–Johnson filtration and [Formula: see text] coincides with the lower central series of [Formula: see text]. In a series of this research, we obtain additional results on the integral (co)homology groups of [Formula: see text]. In particular, we determine the first homology group, and study the cup product of first cohomologies of [Formula: see text]. Furthermore, we construct nontrivial second homology classes of [Formula: see text] by observing its generators and relators, and show that the second cohomology group is not generated by cup products of the first cohomology groups.