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

TL;DR

This paper investigates the finite presentability of certain subgroups of automorphisms of free groups, specifically those containing the IA subgroup, which are analogues of partial Torelli subgroups in mapping class groups.

Contribution

It establishes finite presentability for specific infinite index subgroups of Aut(F_n) that contain IA_n, advancing understanding of their algebraic structure.

Findings

Finite presentability of certain subgroups containing IA_n.

Identification of these subgroups as analogues of partial Torelli groups.

Progress on the open problem for n ≥ 4.

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.