Abstract

We find an upper bound for the asymptotic dimension of a hyperbolic metric space with a set of geodesics satisfying a certain boundedness condition studied by Bowditch. The primary example is a collection of tight geodesics on the curve graph of a compact orientable surface. We use this to conclude that a curve graph has a finite asymptotic dimension. It follows then that a curve graph has property A 1 . We also compute the asymptotic dimension of mapping class groups of orientable surfaces with genus at most 2.