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The asymptotic dimension of a curve graph is finite

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 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 $\le 2$.