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

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 . 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.

Milnor invariants of braids and welded braids up to homotopy

“…We give here a short introduction to the theory of strongly central filtrations and their associated Lie rings. Details may be found in [Dar17,Dar18].…”

“…This was disproved very recently [Bar16], but the methods used do not give a good understanding of what is going on. The Andreadakis equality is known to hold for certain well-behaved subgroups, such as the pure braid group P n [Sat17,Dar18], but the problem stays largely open. In particular, it is open for the group P Σ n of basis-conjugating automorphisms (that is, for the group of pure welded braids), of which our group hP Σ n is a simpler version.…”

“…Proof. We adapt the proof for P n given in [Dar18]. Let w ∈ hP n , and suppose that w acts on RF n as an element of A C j .…”