#### Volume 10, issue 4 (2010)

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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 $4\phantom{\rule{1em}{0ex}}\left(\phantom{\rule{0.3em}{0ex}}mod\phantom{\rule{0.3em}{0ex}}\phantom{\rule{1em}{0ex}}6\right)$. 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

$\underset{g\to \infty }{limsup}\phantom{\rule{1em}{0ex}}{\left({\delta }_{g}\right)}^{g}\le \frac{3+\sqrt{5}}{2}.$

##### Keywords
Teichmüller polynomial, pseudo-Anosov mapping classes, minimal dilatations
Primary: 57M50
Secondary: 57M25