Small asymptotic translation lengths of pseudo-Anosov maps on the curve complex

Abstract

Let $M$ be a hyperbolic fibered 3-manifold whose first Betti number is greater than 1 and let $S$ be a fiber with pseudo-Anosov monodromy $\psi$. We show that there exists a sequence $(R_n,{\psi }_n)$ of fibers and monodromies contained in the fibered cone of $(S,\psi )$ such that the asymptotic translation length of ${\psi }_n$ on the curve complex $\mathcal{C}(R_n)$ behaves asymptotically like 1=j .Rn/j2. As applications, we can reprove the previous result by Gadre–Tsai that the minimal asymptotic translation length of a closed surface of genus g asymptotically behaves like $1/|\chi (R_n){|}^2$. We also show that this holds for the cases of hyperelliptic mapping class group and hyperelliptic handlebody group.