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

Strenner, Balázs

doi:10.2140/agt.2023.23.4087

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

Balázs Strenner

Algebraic & Geometric Topology 23 (2023) 4087–4142

DOI: 10.2140/agt.2023.23.4087

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 $μ_{d}$ for every integer $d \geq 1$ on the rational points of a fibered face of the unit ball of the Thurston norm on $H^{1} (M; ℝ)$ . We show that, even though the functions $μ_{d}$ 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 entropy 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 $μ_{1}$ at infinitely many points for the mapping torus of the simplest pseudo-Anosov braid to show that the values of $μ_{1}$ are rather arbitrary. This suggests that giving a formula for the functions $μ_{d}$ seems very difficult even in the simplest cases.

Keywords

3–manifold, fibrations, veering triangulation

Mathematical Subject Classification

Primary: 57M10, 57M50, 57M60

Secondary: 11H06, 11P21

References

Publication

Received: 14 March 2021

Revised: 30 January 2022

Accepted: 22 February 2022

Published: 23 November 2023

Authors

Balázs Strenner
Department of Mathematics Georgia Institute of Technology Atlanta, GA United States

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Volume 23, issue 9 (2023)