Asymptotic dimension of outer automorphism groups of free groups

Determine the asymptotic dimension of $\operatorname{Out}(F_n)$ for free groups of rank $n\geq3$.

Background

The paper relates projective filling geodesic currents to a broader space of hyperbolic invariant pseudometrics and notes that, for free groups, this broader space contains an isometrically embedded copy of Culler–Vogtmann Outer space.

Although the paper proves infinite rank and asymptotic dimension for the corresponding space associated to a closed surface group, it records that the asymptotic dimension of Out⁡(Fn)\operatorname{Out}(F_n) remains unresolved for n≥3n\geq3.

References

The asymptotic dimension of $\Out(F_n)$ remains unknown for $n\geq 3$.

— Large-scale rank and geometry of the Lipschitz metric on geodesic currents  (2610.03359 - Martínez-Granado et al., 2 Oct 2026) in Section “Further questions,” paragraph “The space of pseudometrics, and free groups”