Pseudo-$F_4$ is isomorphic to $F_4$

Abstract

We prove that the "pseudo-$F_4$" group is isomorphic to $F_4$, answering a question of Brin. Both of these groups can be described as fast groups of homeomorphisms of the interval generated by bumps, as introduced by Bleak, Brin, Kassabov, Moore, and Zaremsky. The proof uses a representation of fast groups as Guba-Sapir diagram groups in order to leverage known results on isomorphisms of diagram groups.