We prove that the braid group B4 on 4 strings, as well as its central
quotient B4/, have the property RD of Haagerup-Jolissaint. It follows
that the automorphism group \Aut(F_2) of the free group F2 on 2 generators
has property RD. We also prove that the braid group B4 is a group of
intermediate rank (of dimension 3). Namely, we show that both B4 and its
central quotient have exponential mesoscopic rank, i.e., that they contain
exponentially many large flat balls which are not included in flats.Comment: reference added, minor correction