Braids, inner automorphisms and the Andreadakis problem

In this paper, we generalize the tools that were introduced in [13] in order to study the Andreadakis problem for subgroups of IA n . In particular, we study the behaviour of the Andreadakis problem when we add inner automorphisms to a subgroup of IA n . We notably use this to show that the Andreadakis equality holds for the pure braid group on n strands modulo its center acting on the free group F n-1 , that is, for the (pure, based) mapping class group of the n-punctured sphere acting on its fundamental group.