On the Andreadakis Problem for Subgroups of IA n

Let Fₙ be the free group on n generators. Consider the group IAₙ of automorphisms of Fₙ acting trivially on its abelianization. There are two canonical filtrations on IAₙ: the 1st one is its lower central series Γ _*; the 2nd one is the Andreadakis filtration A_*, defined from the action on Fₙ. 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.