Let be a geodesic metric space with uniformly generated. If has asymptotic dimension one then 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 and one boundary component is at least two.
"A note on spaces of asymptotic dimension one." Algebr. Geom. Topol. 7 (2) 1063 - 1070, 2007. https://doi.org/10.2140/agt.2007.7.1063