Gregory Bell, Koji Fujiwara
Published 2005-09-09, updated 2007-01-26Version 4
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 $\le 2$.
Comments: 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
Dehn surgeries on the figure eight knot: an upper bound for the complexity
Uniform hyperbolicity of the curve graph via surgery sequences
On Translation Lengths of Pseudo-Anosov Maps on the Curve Graph
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