2021/07/19 by Hyungryul Baik, Baik, Hyungryul, Dongryul M. Kim +3 · 1 citation
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2107.09018
openalex publication_date 2021/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Sg be a closed orientable surface of genus g > 1. Consider the minimal asymptotic translation length LT(k, g) on the Teichmüller space of Sg, among pseudo-Anosov mapping classes of Sg acting trivially on a k-dimensional subspace of H1(Sg), 0 ≤ k ≤ 2g. The asymptotics of LT(k, g) for extreme cases k = 0, 2g have been shown by several authors. Jordan Ellenberg asked whether there is a lower bound for LT(k, g) interpolating the known results on LT(0, g) and LT(2g, g), which was affirmatively answered by Agol, Leininger, and Margalit. In this paper, we study an analogue of Ellenberg's question, replacing Teichmüller spaces with curve complexes. We provide lower and upper bound on the minimal asymptotic translation length LC(k, g) on the curve complex, whose lower bound interpolates the known results on LC(0, g) and LC(2g, g). Finally, for each g, we construct a non-Torelli pseudo-Anosov fg ∈ Mod(Sg) which does not normally generates Mod(Sg) and so that the asymptotic translation length of fg on curve complexes decays more quickly than a constant multiple of 1/g as g → ∞. From this, we provide a restriction on how small the asymptotic translation lengths on curve complexes should be if the similar phenomenon as in the work of Lanier and Margalit on Teichmüller spaces holds for curve complexes.