# Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel–Veech constants

@article{Eskin2002ModuliSO, title={Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel–Veech constants}, author={A. V. Eskin and Howard A. Masur and Anton Zorich}, journal={Publications Math{\'e}matiques de l'Institut des Hautes {\'E}tudes Scientifiques}, year={2002}, volume={97}, pages={61-179} }

A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment (saddle connection) joining a pair of conical points one can find with a nonzero probability another saddle connection on S having the same direction and the same length as the initial one. A similar phenomenon is valid for the families of parallel closed geodesics.We give a complete description of…

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