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

Sylvain Barré (1) , Mikael Pichot (2)

1 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique
2 UTokyo - The University of Tokyo

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.

Domaines: Théorie des groupes [math.GR]; Topologie géométrique [math.GT]; Algèbres d'opérateurs [math.OA]

Sylvain Barré, Mikael Pichot. The 4-string Braid group $B_4$ has property RD and exponential mesoscopic rank. 2009. ⟨hal-00317905v2⟩