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

摘要

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(n)) such that the asymptotic translation length of psi(n) on the curve complex partial derivative(R-n) behaves asymptotically like 1/vertical bar chi(R-n)vertical bar(2) . 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/g(2). We also show that this holds for the cases of hyperelliptic mapping class group and hyperelliptic handlebody group.