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

TL;DR

This work proves ME–superrigidity for Out(F_N) when N≥3, showing that any group measure equivalent to Out(F_N) is virtually isomorphic to it. The authors develop three canonical, Out(F_N)–equivariant splittings attached to subgroups: one from invariant free splittings, one from invariant Z_max splittings via JSJ theory, and a dynamical decomposition based on maximal invariant almost free factor systems. A key technical advance is an improved amenability result for the action of Out(F_N, F) on arational trees with amenable stabilizers, which feeds into a groupoid approach to measure equivalence. The paper then translates these canonical splittings into the language of measured groupoids and cocycles, proving coupling rigidity and orbit equivalence rigidity consequences as well as constraints on lattice embeddings. The results connect the dynamics of free group automorphisms with ergodic and von Neumann algebra rigidity phenomena, advancing the understanding of Out(F_N) as a rigid ME object with strong structural consequences for actions and embeddings.

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.

Figures (9)

• Figure 1: The dynamical decomposition of the group $H\subseteq\mathrm{Out}(F_N)$ induced by homeomorphisms of the three surfaces in Example \ref{['example_intro']}. The conjugacy class of the free factor $Q_1$ is $H$-invariant, and ${\hat{\mathcal{F}}}_1=\{[Q_1],[\langle c_1 \rangle]\}$ is a maximal $H$-invariant almost free factor system.
• Figure 2: Splittings such that $S_{{\mathcal{Z}_{\mathrm{max}}}}$ is trivial.
• Figure 3: The canonical splitting $U^{1}_H$ encoding all $H$-invariant free splittings.
• Figure 4: Example \ref{['ex:arcs']} continued. A maximum $H$-invariant free splitting, dual to the two arcs $\gamma_1,\gamma_2\subseteq \Sigma$.
• Figure 5: A socket graph of groups with 3 proper sockets and 2 improper ones.

Theorems & Definitions (445)

• Definition 1.1: Gromov Gro
• Theorem 1.2: see Theorem \ref{['thm_superrigid']}
• Definition 1.3
• Theorem 1.4: OE-superrigidity, see Theorem \ref{['theo:oe-out(fn)']}
• Theorem 1.5: see proof page \ref{['proof:lattice']}
• Corollary 1.6: see proof page \ref{['proof:lattice']}
• Theorem 1.7: see Theorem \ref{['theo:reducibility']}
• Theorem 1.8: see Theorem \ref{['theo:tree-of-cyl']}
• Theorem 1.9: see Proposition \ref{['prop:chain-FS2']}
• Theorem 1.10: see Theorem \ref{['theo:JSJ_zmax']}