Work

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

Year: 2019

Type: article

Abstract: Let M be a hyperbolic fibered 3-manifold whose first Betti number is greater than 1 and let S be a fiber with pseudo-Anosov monodromy \psi . We show that there exists a sequence (R_n, \psi_n) of fibers and monodromies contained in the fibered cone of (S,\psi) such that the asymptotic translation length of \psi_n on the curve complex \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/|\chi(R_n)|^2 . We also show that this holds for the cases of hyperelliptic mapping class group and hyperelliptic handlebody group.

Source: Groups Geometry and Dynamics

DOI: 10.4171/ggd/508

Authors: Eiko Kin, Hyunshik Shin

Institutions: The University of Osaka, Korea Advanced Institute of Science and Technology, University of Georgia

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Research Foundation, Ministry of Science, ICT and Future Planning, National Research Foundation of Korea, Japan Society for the Promotion of Science

Award: Dynamics of pseudo-Anosov maps and topology of fibered 3-manifolds