The Andreadakis conjecture

Abstract

For $n\ge2$, let $F_n$ be the free group of rank $n$, and let $\IA(F_n)$ be the group of automorphisms of $F_n$ acting trivially on its abelianization. Two natural filtrations of $\IA(F_n)$ are its lower central series $\gamma_c(\IA(F_n))$ and the Andreadakis filtration $\mathrm{I}_{c+1}\mathrm{A}(F_n)$, consisting of automorphisms acting trivially on $F_n/\gamma_{c+1}(F_n)$, $c \geq 1$. Andreadakis conjectured that these filtrations coincide. We disprove this conjecture for every $n\ge3$ and $c\ge4$.