On the stable Andreadakis problem

TL;DR

This paper investigates the relationship between two filtrations on the automorphism group of a free group, establishing stable surjectivity of a key morphism and exploring a p-restricted variant, with additional Lie algebra calculations.

Contribution

It proves the stable surjectivity of the morphism between associated graded Lie rings of two filtrations on IA_n and explores a p-restricted version of the problem.

Findings

The canonical morphism between graded Lie rings is stably surjective.

A p-restricted version of the Andreadakis problem is analyzed.

Lie algebra of the classical congruence group is computed.

Abstract

Let $F\_n$ be the free group on $n$ generators. Consider the group $IA\_n$ of automorpisms of $F\_n$ acting trivially on its abelianization. There are two canonical filtrations on $IA\_n$: the first one is its lower central series $\Gamma\_*$; the second one is the Andreadakis filtration $\mathcal A\_*$, defined from the action on $F\_n$. In this paper, we establish that the canonical morphism between the associated graded Lie rings ${\mathcal L}(\Gamma\_*)$ and ${\mathcal L}(\mathcal A\_*)$ 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.