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

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

2003-08-28

arxiv.org/abs/math/0308268v4

Geom. Topol. Monogr. 7 (2004) 509-547

Mathematics

Geometric Topology

Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon7/paper17.abs.html

Scientific paper

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.

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