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We prove that the braid group $B_4$ on 4 strings, as well as its centralquotient $B_4/< z>$, have the property RD of Haagerup-Jolissaint. It followsthat the automorphism group $\Aut(F_2)$ of the free group $F_2$ on 2 generatorshas property RD. We also prove that the braid group $B_4$ is a group ofintermediate rank (of dimension 3). Namely, we show that both $B_4$ and itscentral quotient have exponential mesoscopic rank, i.e., that they containexponentially many large flat balls which are not included in flats.

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