A note on spaces of asymptotic dimension one

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 $g \geq 2$ and one boundary component is at least two.

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Keywords

asymptotic dimension, quasi-isometry, curve graph

Mathematical Subject Classification 2000

Primary: 51F99

Secondary: 20F69, 54F45, 57M50

References

Publication

Received: 30 November 2006

Revised: 1 May 2007

Accepted: 29 May 2007

Published: 20 June 2007

Authors

Koji Fujiwara
Graduate School of Information Sciences Tohoku University Sendai 980-8579 Japan

Kevin Whyte
Department of Mathematics University of Illinois at Chicago Chicago IL 60607-7045 USA

Volume 7, issue 2 (2007)

Koji Fujiwara and Kevin Whyte

Abstract