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

Hyungryul Baik, Eiko Kin, Hyunshik Shin and Chenxi Wu

Algebraic & Geometric Topology 23 (2023) 1363–1398

Let M be a hyperbolic fibered 3–manifold. We study properties of sequences (Sαn,ψαn) of fibers and monodromies for primitive integral classes in the fibered cone of M. The main object is the asymptotic translation length 𝒞(ψαn) of the pseudo-Anosov monodromy ψαn 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,ψ) in the fibered cone, 𝒞(ψ) is bounded from above by C|χ(S)|. We also obtain a moral connection between 𝒞(ψ) and the normal generating property of ψ in the mapping class group on S. We show that for all but finitely many primitive integral classes (S,ψ) in an arbitrary 2–dimensional slice of the fibered cone, ψ 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.

asymptotic translation length, fibered 3–manifold, fibered cone, pseudo-Anosov, curve complex
Mathematical Subject Classification
Primary: 30F60, 37E30
Secondary: 32G15, 37B40
Received: 28 March 2021
Revised: 10 August 2021
Accepted: 11 October 2021
Published: 6 June 2023
Hyungryul Baik
Department of Mathematical Sciences
South Korea
Eiko Kin
Center for Education in Liberal Arts and Sciences
Osaka University
Hyunshik Shin
Reinsurance Group of America
Chesterfield, MO
United States
Chenxi Wu
Department of Mathematics
University of Wisconsin at Madison
Madison, WI
United States

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