Abstract
We study the combinatorial geometry of “lattice” Jenkins–Strebel differentials with simple zeroes and simple poles on \(\mathbb{C }\!\mathrm{P }^1\) and of the corresponding counting functions. Developing the results of Kontsevich (Commun Math Phys 147:1–23, 1992) we evaluate the leading term of the symmetric polynomial counting the number of such “lattice” Jenkins–Strebel differentials having all zeroes on a single singular layer. This allows us to express the number of general “lattice” Jenkins–Strebel differentials as an appropriate weighted sum over decorated trees. The problem of counting Jenkins–Strebel differentials is equivalent to the problem of counting pillowcase covers, which serve as integer points in appropriate local coordinates on strata of moduli spaces of meromorphic quadratic differentials. This allows us to relate our counting problem to calculations of volumes of these strata . A very explicit expression for the volume of any stratum of meromorphic quadratic differentials recently obtained by the authors (Athreya et al. 2012) leads to an interesting combinatorial identity for our sums over trees.
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Acknowledgments
The authors are happy to thank IHES, IMJ, IUF, MPIM, and the Universities of Chicago, of Illinois at Urbana-Champaign, of Rennes 1, and of Paris 7 for hospitality during the preparation of this paper. We thank the anonymous referee for their careful reading of the paper and helpful suggestions.
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Jayadev S. Athreya is partially supported by NSF grants DMS 0603636, DMS 0244542 and DMS 1069153. Alex Eskin is partially supported by NSF grants DMS 0244542 and DMS 0604251. Anton Zorich is partially supported by the program PICS of CNRS and by ANR “GeoDyM”.
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Athreya, J.S., Eskin, A. & Zorich, A. Counting generalized Jenkins–Strebel differentials. Geom Dedicata 170, 195–217 (2014). https://doi.org/10.1007/s10711-013-9877-7
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DOI: https://doi.org/10.1007/s10711-013-9877-7