Translation distance bounds for fibered 3-manifolds with boundary

2019/02/18 by Alexander J Stas, Stas, Alexander
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1902.06388

openalex publication_date 2019/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given M_φ, a fibered 3-manifold with boundary, we show that the translation distance of the monodromy φ can be bounded above by the complexity of an essential surface with non-zero slope. Furthermore we prove that the minimal complexity of a surface with non-zero slope in M_φⁿ tends to infinity as n→∞. Additionally, we show that an infinite family of fibered hyperbolic knots has translation distance bounded above by two, satisfying a conjecture by Schleimer which postulates that this behavior should hold for all fibered knots.

Citations

• Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle

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