We prove that the full automorphism group and the outer automorphism group of the free group of countably infinite rank are coarsely bounded. That is, these groups admit no continuous actions on a metric space with unbounded orbits and have the quasi-isometry type of a point.

We would like to thank Mladen Bestvina for comments on an earlier version of this paper. We would also like to thank the referee for several helpful comments. The authors acknowledge support from NSF grants DMS-1906095 (Hoganson), DMS-1905720 (Domat, Kwak), RTG DMS-1840190 (Domat, Hoganson), and CAREER DMS-2046889 (Kwak).