Small dilatation mapping classes coming from the simplest hyperbolic braid

By E. Hironaka. 2009

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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 gD2;3;4;5 and 8. We obtain the “Lehmer example” in genus gD 5, and Lanneau and Thiffeault’s conjectural minima in the orientable case for all genus g satisfying gD 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 gD 4;6 or 8. We also prove that if g is the minimum dilatation of pseudo-Anosov mapping classes on a genus g surface, then lim sup g!1 . g/ g 3C p 5

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fac09d8cc3d457ce8d35d78141e3ff2561e5c772