A family of pseudo-Anosov braids with small dilatation

Abstract

This paper describes a family of pseudo-Anosov braids with small dilatation. The smallest dilatations occurring for braids with 3,4 and 5 strands appear in this family. A pseudo-Anosov braid with $2g+1$ strands determines a hyperelliptic mapping class with the same dilatation on a genus–$g$ surface. Penner showed that logarithms of least dilatations of pseudo-Anosov maps on a genus–$g$ surface grow asymptotically with the genus like $1∕g$, and gave explicit examples of mapping classes with dilatations bounded above by $log11∕g$. Bauer later improved this bound to $log6∕g$. The braids in this paper give rise to mapping classes with dilatations bounded above by $log\left(2+\sqrt{3}\right)∕g$. They show that least dilatations for hyperelliptic mapping classes have the same asymptotic behavior as for general mapping classes on genus–$g$ surfaces.