On finite presentability of some partial Torelli subgroups of Aut(F_n)

Abstract

Let $F_n$ be the free group of rank $n$, and let $\rho_{ab}:\mathrm{Aut}(F_n)\to \mathrm{GL}_n(\mathbb Z)$ be the map induced by the natural projection $F_n\to\mathbb Z^n$. It is a long-standing open problem whether the subgroup of $\mathrm{IA}$-automorphisms $\mathrm{IA}_n=\mathrm{Ker}\rho_{ab}$ is finitely presented for $n\geq 4$. In this paper we establish finite presentability of certain infinite index subgroups of $\mathrm{Aut}(F_n)$ containing $\mathrm{IA}_n$. In the terminology of Putman, these subgroups are natural analogues of partial Torelli subgroups of mapping class groups.