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

TL;DR

This paper investigates minimal dilatations of pseudo-Anosov monodromies from the magic 3-manifold, revealing asymptotic bounds and new families of pseudo-Anosovs with orientable invariant foliations.

Contribution

It identifies minimal dilatations for pseudo-Anosovs arising from specific Dehn fillings of the magic 3-manifold and establishes their asymptotic behavior, including new families for certain genera.

Findings

Minimal dilatations achieved by monodromies from specific Dehn fillings.

Asymptotic bounds for dilatations as genus or punctures grow large.

Discovery of new pseudo-Anosov families with orientable invariant foliations.

Abstract

This paper concerns the set $\hat{\mathcal{M}}$ 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. We prove that for each $g$ (resp. $g \not\equiv 0 \pmod{6}$), the minimum among dilatations of elements (resp. elements with orientable invariant foliations) of $\hat{\mathcal{M}}$ defined on a closed surface $\varSigma_g$ of genus $g$ is achieved by the monodromy of some $\varSigma_g$-bundle over the circle obtained from $N(\tfrac{3}{-2})$ or $N(\tfrac{1}{-2})$ by Dehn filling two cusps. These minimizers are the same ones identified by Hironaka, Aaber-Dunfiled, Kin-Takasawa independently. In the case $g \equiv 6 \pmod{12}$ we find a new family of pseudo-Anosovs defined on $\varSigma_g$ with orientable invariant foliations obtained from N(-6) or N(4) by Dehn filling two cusps. We prove that if $\delta_g^+$ is the minimal dilatation of pseudo-Anosovs with orientable invariant foliations defined on $\varSigma_g$, then $$ \limsup_{\substack{g \equiv 6 \pmod{12} g \to \infty}} g \log \delta^+_g \le 2 \log \delta(D_5) \approx 1.0870,$$ where $\delta(D_n)$ 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 $\delta_{1,n}$ is the minimal dilatation of pseudo-Anosovs on a genus 1 surface with $n$ punctures, then $$ \limsup_{n \to \infty} n \log \delta_{1,n} \le 2 \log \delta(D_4) \approx 1.6628. $$