Small dilatation mapping classes coming from the simplest hyperbolic braid

Hironaka, Eriko

doi:10.2140/agt.2010.10.2041

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Small dilatation mapping classes coming from the simplest hyperbolic braid

Eriko Hironaka

Algebraic & Geometric Topology 10 (2010) 2041–2060

DOI: 10.2140/agt.2010.10.2041

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 $δ_{g}$ is the minimum dilatation of pseudo-Anosov mapping classes on a genus $g$ surface, then

\underset{g \to \infty}{limsup} {(δ_{g})}^{g} \leq \frac{3 + \sqrt{5}}{2} .

Keywords

Teichmüller polynomial, pseudo-Anosov mapping classes, minimal dilatations

Mathematical Subject Classification 2000

Primary: 57M50

Secondary: 57M25

References

Publication

Received: 29 October 2009

Revised: 16 April 2010

Accepted: 10 May 2010

Published: 30 September 2010

Authors

Eriko Hironaka
Department of Mathematics Florida State University Tallahasse FL 32301 USA

Volume 10, issue 4 (2010)