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

This paper concerns the set $\hat{ℳ}$ 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 $N (r)$ be the manifold obtained from $N$ by Dehn filling one cusp along the slope $r \in ℚ$ . We prove that for each $g$ (resp. $g ≢ 0 (mod 6)$ ), the minimum among dilatations of elements (resp. elements with orientable invariant foliations) of $\hat{ℳ}$ defined on a closed surface $Σ_{g}$ of genus $g$ is achieved by the monodromy of some $Σ_{g}$ –bundle over the circle obtained from $N (\frac{3}{- 2})$ or $N (\frac{1}{- 2})$ by Dehn filling both cusps. These minimizers are the same ones identified by Hironaka, Aaber and Dunfield, Kin and Takasawa independently. In the case $g \equiv 6 (mod 12)$ we find a new family of pseudo-Anosovs defined on $Σ_{g}$ with orientable invariant foliations obtained from $N (- 6)$ or $N (4)$ by Dehn filling both cusps. We prove that if $δ_{g}^{+}$ is the minimal dilatation of pseudo-Anosovs with orientable invariant foliations defined on $Σ_{g}$ , then

\underset{g \equiv 6 (mod 12), g \to \infty}{limsup} g log δ_{g}^{+} \leq 2 log δ (D_{5}) \approx 1.0870,

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

\underset{n \to \infty}{limsup} n log δ_{1, n} \leq 2 log δ (D_{4}) \approx 1.6628 .

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Keywords

mapping class group, pseudo-Anosov, dilatation, entropy, hyperbolic volume , fibered $3$–manifold, magic manifold

Mathematical Subject Classification 2010

Primary: 57M27, 37E30

Secondary: 37B40

References

Publication

Received: 18 October 2011

Revised: 15 February 2013

Accepted: 19 February 2013

Published: 10 October 2013

Authors

Eiko Kin
Department of Mathematics Graduate School of Science Osaka University Toyonaka Osaka 560-0043 Japan
http://www.math.sci.osaka-u.ac.jp/~kin/
Sadayoshi Kojima
Department of Math. and Computing Sciences Tokyo Institute of Technology Ohokayama, Meguro Tokyo 152-8552 Japan
http://www.is.titech.ac.jp/~sadayosi/
Mitsuhiko Takasawa
Department of Mathematical and Computing Sciences Tokyo Institute of Technology Ohokayama, Meguro Tokyo 152-8552 Japan
http://www.is.titech.ac.jp/~takasawa/

Volume 13, issue 6 (2013)

Eiko Kin, Sadayoshi Kojima and Mitsuhiko Takasawa

Abstract