Andreadakis equality for free-group automorphisms

Determine whether the lower central series terms \(\Gamma_k IA_n\) of the IA-automorphism group \(IA_n=A_{F_n}(1)\) of a free group \(F_n\) of rank \(n\ge 4\) coincide with the Andreadakis–Johnson filtration terms \(A_{F_n}(k)\) for every \(k\ge 1\).

Background

For a group $G$, the Andreadakis–Johnson filtration $A_G(k)$ records the level at which an automorphism acts trivially on the nilpotent quotient $G/\Gamma_{k+1}G$. For the free group $F_n$, its first term $A_{F_n}(1)=IA_n$ is the IA-automorphism group, while $\Gamma_k IA_n$ denotes the $k$th term of the lower central series of $IA_n$. The Andreadakis problem asks whether these two filtrations agree.

The equality is known for all $k$ when $n=2$, for $k=2$ and all $n\ge2$, and for $k=3$ and all $n\ge3$. Bartholdi’s results show that the original conjecture fails for $n=3$, since the quotients $A_{F_3}(4)/\Gamma_4 IA_3$ and $A_{F_3}(5)/\Gamma_5 IA_3$ are nontrivial. The unresolved case explicitly identified by the paper is therefore the equality for free groups of rank $n\ge4$.