#### Volume 7, issue 2 (2007)

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A note on spaces of asymptotic dimension one

### Koji Fujiwara and Kevin Whyte

Algebraic & Geometric Topology 7 (2007) 1063–1070
 arXiv: math.MG/0610391
##### Abstract

Let $X$ be a geodesic metric space with ${H}_{1}\left(X\right)$ uniformly generated. If $X$ has asymptotic dimension one then $X$ is quasi-isometric to an unbounded tree. As a corollary, we show that the asymptotic dimension of the curve graph of a compact, oriented surface with genus $g\ge 2$ and one boundary component is at least two.

##### Keywords
asymptotic dimension, quasi-isometry, curve graph
##### Mathematical Subject Classification 2000
Primary: 51F99
Secondary: 20F69, 54F45, 57M50