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.mod 6/. 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 δg is the minimum dilatation of pseudo-Anosov mapping classes on a genus g surface, then.