  We prove that the braid group $B_4$ on 4 strings, as well as its central
quotient $B_4/< z>$, 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.
