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 $g \equiv 0,1,5,6,7,9 \pmod{10}$ and $\min \Lambda^+_g(-3/2)< \min \Lambda^+_g(-1/2)$ for $g \equiv 1,5,7,9 \pmod{10}$. 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 $\displaystyle \lim_{g \to \infty} |\chi(\Sigma_g)| \log \lambda (\Phi_g(r)) = 2 \log((3+\sqrt{5})/2)$.