New York Journal of Mathematics
Volume 28 (2022), 1506-1511


George Domat, Hannah Hoganson, and Sanghoon Kwak

The automorphism group of the infinite-rank free group is coarsely bounded

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Published: November 18, 2022.
Keywords: Automorphism groups, Out(Fn), coarse geometry, Polish groups.
Subject [2020]: 20E36, 20B27, 22F50, 54H05.

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).

Author information

George Domat:
Department of Mathematics
Rice University
Houston, TX 77005, USA


Hannah Hoganson:
Department of Mathematics
University of Maryland
College Park, MD 20742, USA


Sanghoon Kwak:
Department of Mathematics
University of Utah
Salt Lake City, UT 84112, USA