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

Sylvain Barré (1) , Mikael Pichot (2)

1 LMAM - Laboratoire de Mathématiques et Applications des Mathématiques, EA 3885
2 UTokyo - The University of Tokyo

We prove that the braid group $B_4$ on 4 strings has the property RD of Haagerup-Jolissaint. The case of $B_3$ was resolved by Jolissaint some years ago and the case of higher braid groups is open. We also prove that the braid group $B_4$ is a group of intermediate rank (of dimension 3). Namely, we show that the quotient of $B_4$ by its center is a group of exponential mesoscopic rank. It follows that the braid group $B_4$ itself is of exponential mesoscopic rank, in contrast to the case of the 3-string braid group $B_3$.

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. 2008. ⟨hal-00317905v1⟩