The Andreadakis Problem for the McCool groups
Jaques Darné, Naoya Enomoto, Takao Satoh
TL;DR
This work studies whether the Andreadakis filtration on $\mathrm{Aut}(F_n)$ agrees with the lower central series for the McCool group $\mathrm{P}\Sigma_n$. Using Johnson homomorphisms valued in tangential derivations of the free Lie algebra, the authors construct a concrete counterexample at $n=3$ (a nonzero element $\varpi\in\kappa_6(\mathrm{P}\Sigma_3)$) showing non-injectivity of $\tau_6^{\mathrm{P}\Sigma_3}$ and hence failure of the Andreadakis equality from degree $7$, then propagate this obstruction to all $n\ge3$ via split injections, yielding a central free abelian subgroup of rank $\binom{n}{3}$ in $\big(\mathcal{A}_7(F_n)\cap \mathrm{P}\Sigma_n\big)/\Gamma_7(\mathrm{P}\Sigma_n)$. The paper also analyzes the $\mathfrak{S}_3$-action, computes the representations carried by kernels in low degrees, and provides explicit dimensions up to $k=9$, showing equality holds up to $k\le6$ but not in general. These results demonstrate non-stabilization of the Andreadakis problem for a natural subgroup of IA, highlighting the role of functorial propagation and tangential derivations in automorphism groups of free groups.
Abstract
In this short paper, we show that the McCool group does not satisfy the Andreadakis equality from degree $7$, and we give a lower bound for the size of the difference between the two relevant filtrations. As a consequence, we see that the Andreadakis problem for the McCool group does not stabilize.