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

Let $F_n$ be the free group of rank $n$, and let ${\rho }_{ab}:\mathrm{A}\mathrm{u}\mathrm{t}(F_n)\to {\mathrm{G}\mathrm{L}}_n(\mathbb{Z})$ 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 $\mathrm{I}\mathrm{A}$-automorphisms ${\mathrm{I}\mathrm{A}}_n=\mathrm{K}\mathrm{e}\mathrm{r}{\rho }_{ab}$ is finitely presented for $n\ge 4$. In this paper we establish finite presentability of certain infinite index subgroups of $\mathrm{A}\mathrm{u}\mathrm{t}(F_n)$ containing ${\mathrm{I}\mathrm{A}}_n$. In the terminology of Putman, these subgroups are natural analogues of partial Torelli subgroups of mapping class groups.