A note on spaces of asymptotic dimension one

الأصل · EN

Let X be a geodesic metric space with H₁(X) 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 ≥ 2 and one boundary component is at least two.