Rigidity of the Torelli subgroup in $\mathrm{Out}(F_N)$

Abstract

Let $N\ge 4$. We prove that every injective homomorphism from the Torelli subgroup ${\text{IA}}_N$ to $\text{Out}(F_N)$ differs from the inclusion by a conjugation in $\text{Out}(F_N)$. This applies more generally to the following subgroups of $\text{Out}(F_N)$: every finite-index subgroup of $\text{Out}(F_N)$ (recovering a theorem of Farb and Handel); every subgroup of $\text{Out}(F_N)$ that contains a finite-index subgroup of one of the groups in the Andreadakis–Johnson filtration of $\text{Out}(F_N)$; every subgroup that contains a power of every linearly-growing automorphism; more generally, every twist-rich subgroup of $\text{Out}(F_N)$ –those are subgroups that contain sufficiently many twists in an appropriate sense.
Among applications, this recovers the fact that the abstract commensurator of every group above is equal to its relative commensurator in $\text{Out}(F_N)$; it also implies that all subgroups in the Andreadakis–Johnson filtration of $\text{Out}(F_N)$ are co-Hopfian.
We also prove the same rigidity statement for subgroups of $\text{Out}(F_3)$ which contain a power of every Nielsen transformation. This shows, in particular, that $\text{Out}(F_3)$ and all its finite-index subgroups are co-Hopfian, extending a theorem of Farb and Handel to the $N=3$ case.