A family of pseudo-Anosov braids with small dilatation

TL;DR

This paper introduces a new family of pseudo-Anosov braids with minimal dilatations for small numbers of strands, providing explicit examples and analyzing their asymptotic behavior on surfaces of genus g.

Contribution

It presents a novel family of pseudo-Anosov braids with small dilatations, including the smallest known for 3, 4, and 5 strands, and relates these to hyperelliptic mapping classes.

Findings

The smallest dilatations for 3, 4, and 5 strands are in this family.

Dilatations of hyperelliptic mapping classes grow like 1/g with genus g.

The new braids give upper bounds on dilatations of the form log(2+sqrt(3))/g.

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 log 11/g. Bauer later improved this bound to log 6/g. The braids in this paper give rise to mapping classes with dilatations bounded above by log(2+sqrt(3))/g. They show that least dilatations for hyperelliptic mapping classes have the same asymptotic behavior as for general mapping classes on genus-g surfaces.