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 [math] strands determines a hyperelliptic mapping class with the same dilatation on a genus– [math] surface. Penner showed that logarithms of least dilatations of pseudo-Anosov maps on a genus– [math] surface grow asymptotically with the genus like [math] , and gave explicit examples of mapping classes with dilatations bounded above by [math] . Bauer later improved this bound to [math] . The braids in this paper give rise to mapping classes with dilatations bounded above by [math] . They show that least dilatations for hyperelliptic mapping classes have the same asymptotic behavior as for general mapping classes on genus– [math] surfaces.