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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

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 μd for every integer d 1 on the rational points of a fibered face of the unit ball of the Thurston norm on H1(M; ). We show that, even though the functions μd 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 μ1 at infinitely many points for the mapping torus of the simplest pseudo-Anosov braid to show that the values of μ1 are rather arbitrary. This suggests that giving a formula for the functions μd seems very difficult even in the simplest cases.

3–manifold, fibrations, veering triangulation
Mathematical Subject Classification
Primary: 57M10, 57M50, 57M60
Secondary: 11H06, 11P21
Received: 14 March 2021
Revised: 30 January 2022
Accepted: 22 February 2022
Published: 23 November 2023
Balázs Strenner
Department of Mathematics
Georgia Institute of Technology
Atlanta, GA
United States

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