Measure equivalence rigidity of $\mathrm{Out}(F_N)$

Abstract

We prove that for every $N\ge 3$, the group $\mathrm{Out}(F_N)$ of outer automorphisms of a free group of rank $N$ is superrigid from the point of view of measure equivalence: any countable group that is measure equivalent to $\mathrm{Out}(F_N)$, is in fact virtually isomorphic to $\mathrm{Out}(F_N)$. We introduce three new constructions of canonical splittings associated to a subgroup of $\mathrm{Out}(F_N)$ of independent interest. They encode respectively the collection of invariant free splittings, invariant cyclic splittings, and maximal invariant free factor systems. Our proof also relies on the following improvement of an amenability result by Bestvina and the authors: given a free factor system $\mathcal{F}$ of $F_N$, the action of $\mathrm{Out}(F_N,\mathcal{F})$ (the subgroup of $\mathrm{Out}(F_N)$ that preserves $\mathcal{F}$) on the space of relatively arational trees with amenable stabilizer is a Borel amenable action.