Rigidity of the Torelli subgroup in $Out(F_N)$

TL;DR

This paper proves that the Torelli subgroup and many related subgroups of $Out(F_N)$ exhibit rigidity, meaning injective homomorphisms are essentially conjugations, leading to results on their automorphism structures and co-Hopfian properties.

Contribution

It establishes a broad rigidity theorem for the Torelli subgroup and related subgroups in $Out(F_N)$, extending previous results to the case $N=3$ and beyond.

Findings

Injective homomorphisms differ from inclusions by conjugation.

Subgroups in the Andreadakis--Johnson filtration are co-Hopfian.

The automorphism group of these subgroups is essentially trivial.

Abstract

Let $N$ be at least 4. We prove that every injective homomorphism from the Torelli subgroup into $Out(F_N)$ differs from the inclusion by a conjugation in $Out(F_N)$. This applies more generally to the following subgroups: every finite-index subgroup of $Out(F_N)$ (recovering a theorem of Farb and Handel); every subgroup that contains a finite-index subgroup of one of the groups in the Andreadakis--Johnson filtration; every subgroup that contains a power of every linearly-growing automorphism; more generally, every twist-rich subgroup (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 $Out(F_N)$; it also implies that all subgroups in the Andreadakis--Johnson filtration are co-Hopfian. We also prove the same rigidity statement for subgroups of $Out(F_3)$ which contain a power of every Nielsen transformation. This shows in particular that $Out(F_3)$ and all its finite-index subgroups are co-Hopfian, extending a theorem of Farb and Handel to the $N=3$ case.