Moduli spaces of abelian differentials : the principal boundary, counting problems, and the Siegel-Veech constants
Eskin, Alex ; Masur, Howard ; Zorich, Anton
Publications Mathématiques de l'IHÉS, Tome 98 (2003), p. 61-179 / Harvested from Numdam
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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 all possible configurations of parallel saddle connections (and of families of parallel closed geodesics) which might be found on a generic flat surface S. We count the number of saddle connections of length less than L on a generic flat surface S; we also count the number of admissible configurations of pairs (triples,...) of saddle connections; we count the analogous numbers of configurations of families of closed geodesics. By the previous result of [EMa] these numbers have quadratic asymptotics c·(πL 2 ). Here we explicitly compute the constant c for a configuration of every type. The constant c is found from a Siegel-Veech formula. To perform this computation we elaborate the detailed description of the principal part of the boundary of the moduli space of holomorphic 1-forms and we find the numerical value of the normalized volume of the tubular neighborhood of the boundary. We use this for evaluation of integrals over the moduli space.

@article{PMIHES_2003__97__61_0,
     author = {Eskin, Alex and Masur, Howard and Zorich, Anton},
     title = {Moduli spaces of abelian differentials : the principal boundary, counting problems, and the Siegel-Veech constants},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     volume = {98},
     year = {2003},
     pages = {61-179},
     doi = {10.1007/s10240-003-0015-1},
     zbl = {1037.32013},
     language = {en},
     url = {http://dml.mathdoc.fr/item/PMIHES_2003__97__61_0}
}

Eskin, Alex; Masur, Howard; Zorich, Anton. Moduli spaces of abelian differentials : the principal boundary, counting problems, and the Siegel-Veech constants. Publications Mathématiques de l'IHÉS, Tome 98 (2003) pp. 61-179. doi : 10.1007/s10240-003-0015-1. http://gdmltest.u-ga.fr/item/PMIHES_2003__97__61_0/

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