#### Volume 13, issue 6 (2013)

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Minimal dilatations of pseudo-Anosovs generated by the magic $3$–manifold and their asymptotic behavior

### Eiko Kin, Sadayoshi Kojima and Mitsuhiko Takasawa

Algebraic & Geometric Topology 13 (2013) 3537–3602
##### Abstract

This paper concerns the set $\stackrel{̂}{\mathsc{ℳ}}$ of pseudo-Anosovs which occur as monodromies of fibrations on manifolds obtained from the magic $3$–manifold $N$ by Dehn filling three cusps with a mild restriction. Let $N\left(r\right)$ be the manifold obtained from $N$ by Dehn filling one cusp along the slope $r\in ℚ$. We prove that for each $g$ (resp. $g\not\equiv 0\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}6\right)$), the minimum among dilatations of elements (resp. elements with orientable invariant foliations) of $\stackrel{̂}{\mathsc{ℳ}}$ defined on a closed surface ${\Sigma }_{g}$ of genus $g$ is achieved by the monodromy of some ${\Sigma }_{g}$–bundle over the circle obtained from $N\left(\frac{3}{-2}\right)$ or $N\left(\frac{1}{-2}\right)$ by Dehn filling both cusps. These minimizers are the same ones identified by Hironaka, Aaber and Dunfield, Kin and Takasawa independently. In the case $g\equiv 6\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}12\right)$ we find a new family of pseudo-Anosovs defined on ${\Sigma }_{g}$ with orientable invariant foliations obtained from $N\left(-6\right)$ or $N\left(4\right)$ by Dehn filling both cusps. We prove that if ${\delta }_{g}^{+}$ is the minimal dilatation of pseudo-Anosovs with orientable invariant foliations defined on ${\Sigma }_{g}$, then

$\underset{g\equiv 6\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}12\right),g\to \infty }{limsup}glog{\delta }_{g}^{+}\le 2log\delta \left({D}_{5}\right)\approx 1.0870,$

where $\delta \left({D}_{n}\right)$ is the minimal dilatation of pseudo-Anosovs on an $n$–punctured disk. We also study monodromies of fibrations on $N\left(1\right)$. We prove that if ${\delta }_{1,n}$ is the minimal dilatation of pseudo-Anosovs on a genus $1$ surface with $n$ punctures, then

$\underset{n\to \infty }{limsup}nlog{\delta }_{1,n}\le 2log\delta \left({D}_{4}\right)\approx 1.6628.$

##### Keywords
mapping class group, pseudo-Anosov, dilatation, entropy, hyperbolic volume , fibered $3$–manifold, magic manifold
##### Mathematical Subject Classification 2010
Primary: 57M27, 37E30
Secondary: 37B40