Small asymptotic translation lengths of pseudo-Anosov maps on the curve complex

  • Eiko Kin

    Osaka University, Japan
  • Hyunshik Shin

    KAIST, Daejeon, Republic of Korea and University of Georgia, Athens, USA
Small asymptotic translation lengths of pseudo-Anosov maps on the curve complex cover
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Abstract

Let MM be a hyperbolic fibered 3-manifold whose first Betti number is greater than 1 and let SS be a fiber with pseudo-Anosov monodromy ψ\psi. We show that there exists a sequence (Rn,ψn)(R_n, \psi_n) of fibers and monodromies contained in the fibered cone of (S,ψ)(S,\psi) such that the asymptotic translation length of ψn\psi_n on the curve complex C(Rn)\mathcal C(R_n) behaves asymptotically like 1=j .Rn/j2. As applications, we can reprove the previous result by Gadre–Tsai that the minimal asymptotic translation length of a closed surface of genus g asymptotically behaves like 1/χ(Rn)21/|\chi(R_n)|^2. We also show that this holds for the cases of hyperelliptic mapping class group and hyperelliptic handlebody group.

Cite this article

Eiko Kin, Hyunshik Shin, Small asymptotic translation lengths of pseudo-Anosov maps on the curve complex. Groups Geom. Dyn. 13 (2019), no. 3, pp. 883–907

DOI 10.4171/GGD/508