Immersions of Pants into a Fixed Hyperbolic Surface
Lewis Bowen
Abstract
Exploiting a relationship between closed geodesics on a generic closed hyperbolic surface S and a certain unipotent flow on the product space T1(S) x T1(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 gamma1, gamma2 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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