On the Andreadakis problem for subgroups of $IA\_n$

TL;DR

This paper investigates the relationship between two canonical filtrations on the automorphism group of a free group, showing they coincide in specific subgroups, thus advancing understanding of the Andreadakis problem.

Contribution

It demonstrates that the lower central series and the Andreadakis filtration coincide for triangular automorphisms and pure braid groups.

Findings

Filtrations coincide for triangular automorphisms.

Filtrations coincide for pure braid groups.

Advances understanding of the 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 $\Gamma\_*$; the second one is the Andreadakis filtration $\mathcal A\_*$, defined from the action on $F\_n$. The Andreadakis problem consists in understanding the difference between these filtrations. Here, we show that they coincide when restricted to the subgroup of triangular automorphisms, and to the pure braid group.