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

TL;DR

This paper proves that the braid group B4 and its central quotient have property RD, and introduces the concept of exponential mesoscopic rank, showing B4 has intermediate rank with many large flat balls.

Contribution

It establishes property RD for B4 and its quotient, and introduces exponential mesoscopic rank, revealing B4's intermediate rank and geometric complexity.

Findings

B4 and its quotient have property RD.

B4 exhibits exponential mesoscopic rank.

B4 has intermediate rank with many large flat balls.

Abstract

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.