Geometry & Topology Monographs, Vol. 7 (2004),
Proceedings of the Casson Fest,
Paper no. 17, pages 509--547.

The metric space of geodesic laminations on a surface II: small surfaces

Francis Bonahon, Xiaodong Zhu

Abstract. We continue our investigation of the space of geodesic laminations on a surface, endowed with the Hausdorff topology. We determine the topology of this space for the once-punctured torus and the 4-times-punctured sphere. For these two surfaces, we also compute the Hausdorff dimension of the space of geodesic laminations, when it is endowed with the natural metric which, for small distances, is -1 over the logarithm of the Hausdorff metric. The key ingredient is an estimate of the Hausdorff metric between two simple closed geodesics in terms of their respective slopes.

Keywords. Geodesic lamination, simple closed curve

AMS subject classification. Primary: 57M99, 37E35.

E-print: arXiv:math.GT/0308268

Submitted to GT on 6 October 2003. (Revised 21 April 2005.) Paper accepted 21 April 2005. Paper published 21 May 2005.

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Francis Bonahon, Xiaodong Zhu
Department of Mathematics, University of Southern California
Los Angeles, CA 90089-2532, USA
Juniper Networks, 1194 North Mathilda Avenue
Sunnyvale, CA 94089-1206, USA

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