Pseudo-Anosovs on closed surfaces having small entropy and the Whitehead sister link exterior

Abstract

Let ${\delta }_g$ be the minimal dilatation for pseudo-Anosovs on a closed surface ${\Sigma }_g$ of genus $g$ and let ${\delta }_g^{+}$ be the minimal dilatation for pseudo-Anosovs on ${\Sigma }_g$ with orientable invariant foliations. This paper concerns the pseudo-Anosovs which occur as the monodromies on closed fibers for Dehn fillings of $N(r)$ for each $r\in \{-3/2,-1/2,2\}$ of the magic manifold $N$. The manifold $N(-3/2)$ is homeomorphic to the Whitehead sister link exterior. We consider the set ${\Lambda }_g(r)$ (resp. ${\Lambda }_g^{+}(r)$) which consists of the dilatations of all monodromies (resp. monodromies having orientable invariant foliations) on a closed fiber of genus $g$ for Dehn fillings of $N(r)$, where the fillings are on the boundary slopes of fibers of $N(r)$. Hironaka obtained upper bounds of ${\delta }_g$ and ${\delta }_g^{+}$ by computing $\min {\Lambda }_g(-1/2)$ and $\min {\Lambda }_g^{+}(-1/2)$ respectively. We prove that $\min {\Lambda }_g(-3/2)<\min {\Lambda }_g(-1/2)$ for $g\equiv 0,1,5,6,7,9(\mathrm{mod}10)$ and $\min {\Lambda }_g^{+}(-3/2)<\min {\Lambda }_g^{+}(-1/2)$ for $g\equiv 1,5,7,9(\mathrm{mod}10)$. These inequalities improve the previous upper bounds of ${\delta }_g$ and ${\delta }_g^{+}$ for these $g$. We prove that for each $r\in \{-3/2,-1/2,2\}$ and each $g\ge 3$, there exists a monodromy ${\Phi }_g(r)$ on a closed fiber of genus $g$ for a Dehn filling of $N(r)$ such that its dilatation $\lambda ({\Phi }_g(r))$ satisfies $\underset{g\to \infty }{\lim }|\chi ({\Sigma }_g)|\log \lambda ({\Phi }_g(r))=2\log ((3+\sqrt{5})/2)$.

24 pages, 7 figures; v3: minor modification

References in corpus (3)

• A family of pseudo-Anosov braids with small dilatation
• Small dilatation pseudo-Anosov mapping classes coming from the simplest hyperbolic braid
• Minimal dilatations of pseudo-Anosovs generated by the magic 3-manifold and their asymptotic behavior

Cited by in corpus (5)

• Small dilatation pseudo-Anosov mapping classes coming from the simplest hyperbolic braid
• Minimal dilatations of pseudo-Anosovs generated by the magic 3-manifold and their asymptotic behavior
• On the number and location of short geodesics in moduli space
• Pseudo-Anosov homeomorphisms on translation surfaces in hyperelliptic components have large entropy
• On pseudo-Anosov mapping classes with minimum dilatation and Lanneau-Thiffeault numbers