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Journal of the London Mathematical Society Advance Access originally published online on November 19, 2007
Journal of the London Mathematical Society 2008 77(1):33-50; doi:10.1112/jlms/jdm090
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© 2007 London Mathematical Society

The asymptotic dimension of a curve graph is finite

Gregory C. Bell

Department of Mathematics and Statistics
UNC Greensboro
Greensboro, NC 27455
USA

Koji Fujiwara

Mathematical Institute
Tohoku University
Sendai 980-8578
Japan
fujiwara{at}math.tohoku.ac.jp

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 A1. We also compute the asymptotic dimension of mapping class groups of orientable surfaces with genus at most 2.

Dedicated to Professor Yukio Matsumoto on his 60th birthday

2000 Mathematics Subject Classification 57M99 (primary), 20F69 (secondary).

Received September 30, 2005; revised January 30, 2007; published online November 19, 2007.


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