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 strands determines a hyperelliptic mapping class with the same dilatation on a genus– surface. Penner showed that logarithms of least dilatations of pseudo-Anosov maps on a genus– surface grow asymptotically with the genus like , and gave explicit examples of mapping classes with dilatations bounded above by . Bauer later improved this bound to . The braids in this paper give rise to mapping classes with dilatations bounded above by . They show that least dilatations for hyperelliptic mapping classes have the same asymptotic behavior as for general mapping classes on genus– surfaces.
"A family of pseudo-Anosov braids with small dilatation." Algebr. Geom. Topol. 6 (2) 699 - 738, 2006. https://doi.org/10.2140/agt.2006.6.699