Abstract: The curve complex C(S) of a closed orientable surface S of genus
g is an infinite graph with vertices as isotopy classes of essential simple
closed curves on S with two vertices adjacent by an edge if the curves can
be isotoped to be disjoint. By a celebrated theorem of Masur-Minsky, the
curve complex is Gromov hyperbolic. Moreover, a pseudo-Anosov map f of S
acts on C(S) as a hyperbolic isometry with an invariant quasi-axis. This
allows one to define an asymptotic translation length for f on C(S). In
joint work with Chia-yen Tsai, we prove that the minimal pseudo-Anosov
asymptotic translation lengths on C(S) are of the order 1/g^2. We shall
also outline related interesting results and questions.