Braids, inner automorphisms and the Andreadakis problem

TL;DR

This paper explores the behavior of the Andreadakis problem in relation to subgroups of automorphisms of free groups, especially when inner automorphisms are included, and proves the equality for a specific braid group case.

Contribution

It generalizes previous tools to analyze the Andreadakis problem with inner automorphisms and establishes the equality for a particular pure braid group scenario.

Findings

Andreadakis equality holds for the pure braid group on n strands modulo its center.

Extended the analysis of the Andreadakis problem to include inner automorphisms.

Provided new insights into the structure of subgroups of automorphisms of free groups.

Abstract

In this paper, we generalize the tools that were introduced in [Dar19b] in order to study the Andreadakis problem for subgroups of IAn. In particular, we study the behaviour of the Andreadakis problem when we add inner automorphisms to a subgroup of IAn. 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 on n-1 generators , that is, for the (pure, based) mapping class group of the n-punctured sphere acting on its fundamental group.