The 4-string braid group $B_4$ has property RD and exponential mesoscopic rank

Abstract

We prove that the braid group $B_4$ on 4 strings, as well as its central quotient $B_4/\langle z\rangle$, have the property RD of Haagerup-Jolissaint. It follows that the automorphism group $\Aut(F_2)$ of the free group $F_2$ on 2 generators has property RD. We also prove that the braid group $B_4$ is a group of intermediate rank (of dimension 3). Namely, we show that both $B_4$ and its central quotient have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.