Small dilatation mapping classes coming from the simplest hyperbolic braid

Abstract

In this paper we study the small dilatation pseudo-Anosov mapping classes arising from fibrations over the circle of a single 3–manifold, the mapping torus for the “simplest hyperbolic braid”. The dilatations that occur include the minimum dilatations for orientable pseudo-Anosov mapping classes for genus $g=2,3,4,5$ and $8$. We obtain the “Lehmer example” in genus $g=5$, and Lanneau and Thiffeault’s conjectural minima in the orientable case for all genus $g$ satisfying $g=2$ or $4\left(mod6\right)$. Our examples show that the minimum dilatation for orientable mapping classes is strictly greater than the minimum dilatation for non-orientable ones when $g=4,6$ or $8$. We also prove that if ${\delta }_g$ is the minimum dilatation of pseudo-Anosov mapping classes on a genus $g$ surface, then

$$\underset{g\to \infty }{limsup}{\left({\delta }_g\right)}^g\le \frac{3+\sqrt{5}}{2}.$$