Algebra/Topology seminar – University of Copenhagen

Algebra/Topology seminar

Jacques Darne (Lille), Studying 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. Although an explicit finite set of generators of this group has been known for a long time, the structure of $IA_n$ remains largely mysterious. For instance, $IA_3$ is not finitely presented, and it is not known if $IA_n$ is finitely presented for $n > 3$. To study its structure, we can define 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 $F_n$. In the 1970s, Andreadakis conjectured that they were the same, but recent calculations disproved this conjecture. In this talk, I will describe several variations on this problem and will produce some (partial) answers.