Experiments Suggesting That the Distribution of the Hyperbolic Length of Closed Geodesics Sampling by Word Length Is Gaussian

Abstract

Each free homotopy class of directed closed curves on a surface with boundary can be described by a cyclic reduced word in the generators of the fundamental group and their inverses. The word length is the number of letters of the cyclic word. If the surface has a hyperbolic metric with geodesic boundary, the geometric length of the class is the length of the unique geodesic in that class. By computer experiments, we investigate the distribution of the geometric length among all classes with a given word length on the pair of pants surface. Our experiments strongly suggest that the distribution is normal.

2000 AMS Subject Classification:

• 57M50

Keywords:

• hyperbolic surfaces
• geodesics
• central limit theorem
• word length

Acknowledgments

This work was supported by NSF grant 1098079/1/58949.

Notes

¹The data on which the histograms are based can be found at http://www.math.sunysb.edu/moira/HyperbolicLengthData/ .