On the stable Andreadakis problem

Abstract

Let F n be the free group on n generators. Consider the group IA n of automorphisms of F n acting trivially on its abelianization. There are two canonical filtrations on IA n : the first one is its lower central series Γ * ; the second one is the Andreadakis filtration A * , defined from the action on F n . In this paper, we establish that the canonical morphism between the associated graded Lie rings L(Γ * ) and L(A * ) is stably surjective. We then investigate a p-restricted version of the Andreadakis problem. A calculation of the Lie algebra of the classical congruence group is also included. arXiv:1711.05991v2 [math.AT]

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

“…We also remark that affirmative answers to the Andreadakis conjecture restricted to certain subgroups of Aut F n were given by [41,43]. These works were systematically generalized by recent notable works of Darné [12,13].…”

The third subgroup of the Andreadakis–Johnson filtration of the automorphism group of a free group

“…For a general n 4, the conjecture is still open. Here we should note a remarkable result due to Darné [9]. Recently, he proved that the stable Andreadakis conjecture is true.…”

Lower central series and split extensions

“…The following "Hall-Witt" identities then hold: rx, yzs " rx, ys ¨yrx, zs " rx, ysrx, zsrrz, xs, ys, (8) rrx, ys, y zsrry, zs, z xsrrz, xs, x ys " 1. (9) Given subgroups H 1 and H 2 of G, define their commutator, rH 1 , H 2 s, to be the subgroup of G generated by all elements of the form rx 1 , x 2 s with x i P H i . In particular, G 1 " rG, Gs is the derived (or, commutator) subgroup of G.…”