The asymptotic dimension of a curve graph is finite

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 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 $\le 2$.

19 pages. Made some minor revisions. The section on mapping class groups has been rewritten; in particular we compute the asdim of Mod(S) where S has genus at most 2. The last section on open questions has been modified to reflect recent developments. References have been updated

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