Immersions of Pants into a Fixed Hyperbolic Surface

Details Immersions of Pants into a Fixed Hyperbolic Surface Immersions of Pants into a Fixed Hyperbolic Surface

2005-05-23

arxiv.org/abs/math/0505480v2

Mathematics

Geometric Topology

43 pages

Scientific paper

Exploiting a relationship between closed geodesics on a generic closed hyperbolic surface S and a certain unipotent flow on the product space T_1(S) x T_1(S), we obtain a local asymptotic equidistribution result for long closed geodesics on S. Applications include asymptotic estimates for the number of pants immersions into S satisfying various geometric constraints. Also we show that two uniformly random closed geodesics gamma_1, gamma_2 of length within epsilon of L have a high probability of partially bounding an immersed 4-holed sphere whose other boundary components also have length within epsilon of L. Version 2: The coefficient in corollary 1.6 corrected from 16 to 8.

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