A note on spaces of asymptotic dimension one

Abstract

Let $X$ be a geodesic metric space with $H_1(X)$ 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 $g\ge 2$ and one boundary component is at least two.

add Example 0.3, 7 pages

References in corpus (2)

• Cohomological approach to asymptotic dimension
• The end of the curve complex

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