Let be a hyperbolic fibered 3-manifold whose first Betti number is greater than 1 and let be a fiber with pseudo-Anosov monodromy . We show that there exists a sequence of fibers and monodromies contained in the fibered cone of such that the asymptotic translation length of on the curve complex 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 . 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. 883–907DOI 10.4171/GGD/508