Measure bound for translation surfaces with short saddle connections
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Publication:6335424
DOI10.1007/S00039-023-00636-9zbMath1517.32025arXiv2002.10026MaRDI QIDQ6335424
Publication date: 23 February 2020
Abstract: We prove that any ergodic $SL_2(R)$-invariant probability measure on a stratum of translation surfaces satisfies strong regularity: the measure of the set of surfaces with two non-parallel saddle connections of length at most $epsilon_1, epsilon_2$ is $O(epsilon_1^2 epsilon_2^2)$. We prove a more general theorem which works for any number of short saddle connections. The proof uses the multi-scale compactification of strata recently introduced by Bainbridge-Chen-Gendron-Grushevsky-M"oller and the algebraicity result of Filip.
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