The Automorphism Group of the Infinite-Rank Free Group is Coarsely Bounded

Abstract

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.Comment: 4 pages. v2: Incorporated referee's comment. To appear in New York Journal of Mathematic