Volume 10, issue 4 (2010)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Small dilatation mapping classes coming from the simplest hyperbolic braid

Eriko Hironaka

Algebraic & Geometric Topology 10 (2010) 2041–2060
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(mod6). 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

limsupg(δg)g 3 + 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