Measure bound for translation surfaces with short saddle connections

Abstract

We prove that any ergodic $S{L}_2(R)$-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-Möller and the algebraicity result of Filip.

arXiv comment57 pages, 9 figures. Added Remark 1.5 concerning the opposite inequality. Various minor changes throughout. To appear in GAFA

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• Equations of linear subvarieties of strata of differentials