Asymptotic translation lengths and normal generation for pseudo-Anosov monodromies of fibered 3-manifolds

TL;DR

This paper investigates the asymptotic translation lengths of pseudo-Anosov monodromies in hyperbolic fibered 3-manifolds, establishing bounds, normal generation properties, and the non-existence of continuous extensions of these lengths on the fibered face.

Contribution

It provides bounds on translation lengths, links these lengths to normal generation in the mapping class group, and shows the non-existence of a continuous extension of normalized translation lengths on the fibered face.

Findings

Existence of a uniform bound on translation lengths depending on Euler characteristic

Most primitive classes in the fibered cone have monodromies that normally generate the mapping class group

No continuous extension of normalized translation lengths on the curve complex exists in general

Abstract

Let $M$ be a hyperbolic fibered 3-manifold. We study properties of sequences $(S_{\alpha_n}, \psi_{\alpha_n})$ of fibers and monodromies for primitive integral classes in the fibered cone of $M$. The main tool is the asymptotic translation length $\ell_{\mathcal{C}} (\psi_{\alpha_n})$ of the pseudo-Anosov monodromy $ \psi_{\alpha_n}$ on the curve complex. We first show that there exists a constant $C>0$ depending only on the fibered cone such that for any primitive integral class $(S, \psi)$ in the fibered cone, $\ell_{\mathcal{C}} (\psi)$ is bounded from above by $C/|\chi(S)|$. We also obtain a moral connection between $\ell_{\mathcal{C}} (\psi)$ and the normal generating property of $\psi$ in the mapping class group on $S$. We show that for all but finitely many primitive integral classes $(S, \psi)$ in an arbitrary 2-dimensional slice of the fibered cone, $\psi$ normally generates the mapping class group on $S$. In the second half of the paper, we study if it is possible to obtain a continuous extension of normalized asymptotic translation lengths on the curve complex as a function on the fibered face. An analogous question for normalized entropy has been answered affirmatively by Fried and the question for normalized asymptotic translation length on the arc complex in the fully punctured case has been answered negatively by Strenner. We show that such an extension in the case of the curve complex does not exist in general by explicit computation for sequences in the fibered cone of the magic manifold.