Fibrations of 3-manifolds and asymptotic translation length in the arc complex

TL;DR

This paper studies how the asymptotic translation lengths of pseudo-Anosov monodromies in the arc complex vary across different fibrations of 3-manifolds, revealing complex behavior and dependence on geometric slices.

Contribution

It introduces normalized asymptotic translation length functions on fibered faces, analyzes their accumulation points, and provides explicit descriptions for certain slices, advancing understanding of monodromy dynamics.

Findings

Sets of accumulation points are well-behaved and depend only on the shape of slices.

The functions  are typically nowhere continuous.

Values of  can be arbitrary, indicating complexity in their formulas.

Abstract

Given a 3-manifold $M$ fibering over the circle, we investigate how the asymptotic translation lengths of pseudo-Anosov monodromies in the arc complex vary as we vary the fibration. We formalize this problem by defining normalized asymptotic translation length functions $\mu_d$ for every integer $d \ge 1$ on the rational points of a fibered face of the unit ball of the Thurston norm on $H^1(M;\mathbb{R})$. We show that even though the functions $\mu_d$ themselves are typically nowhere continuous, the sets of accumulation points of their graphs on $d$-dimensional slices of the fibered face are rather nice and in a way reminiscent of Fried's convex and continuous normalized stretch factor function. We also show that these sets of accumulation points depend only on the shape of the corresponding slice. We obtain a particularly concrete description of these sets when the slice is a simplex. We also compute $\mu_1$ at infinitely many points for the mapping torus of the simplest hyperbolic braid to show that the values of $\mu_1$ are rather arbitrary. This suggests that giving a formula for the functions $\mu_d$ seems very difficult even in the simplest cases.