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

TL;DR

This paper demonstrates that for certain hyperbolic fibered 3-manifolds, the asymptotic translation lengths of pseudo-Anosov monodromies on the curve complex decrease quadratically with the Euler characteristic, extending previous results.

Contribution

It constructs a sequence of fibers with monodromies whose translation lengths decay like the inverse square of their Euler characteristics, generalizing prior findings.

Findings

Asymptotic translation length behaves like 1/|χ(R)|^2

Reproves Gadre--Tsai's result for minimal translation length

Extends results to hyperelliptic mapping class and handlebody groups

Abstract

Let $M$ be a hyperbolic fibered 3-manifold with $b_1(M) \geq 2$ 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 $1/|\chi(R_n)|^2$. 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 $1/g^2$. We also show that this also holds for the cases of hyperelliptic mapping class group and hyperelliptic handlebody group.