Square-tiled surfaces of fixed combinatorial type: equidistribution, counting, volumes of the ambient strata
speaker: Anton Zorich (Institut de Mathématiques de Jussieu, France)abstract: A. Zorich, joint work with V.Delecroix, E.Goujard, P.Zograf We prove that square-tiled surfaces (correspondingly pillowcase covers) tiled with tiny squares sharing a fixed combinatorics of cylinder gluing are asymptotically equidistributed in the ambient stratum in the moduli space of Abelian (correspondingly quadratic) differentials. We prove similar equidistribution results for rational interval exchange transformation. We compute explicitly the contribution of square-tiled surfaces (correspondingly pillowcase covers) having a single horizontal cylinder to the volume of the corresponding stratum. The resulting count is particularly simple and efficient in large genus asymptotics. We conjecture that this contribution is asymptotically of the order $1d$ where $d$ is the dimension of the stratum and prove that this conjecture is equivalent to the long-standing conjecture on asymptotics for the volumes of the moduli spaces of Abelian differentials. In certain particular cases the conjecture was recently proved by D. Chen, M. Moller, and D. Zagier.
timetable:
Thu 7 Apr, 9:00 - 10:00, Aula Dini
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