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arXiv · 2002.10026

Measure bound for translation surfaces with short saddle connections

Benjamin Dozier
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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 $ε_1, ε_2$ is $O(ε_1^2 ε_2^2)$. 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.

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Benjamin Dozier. 2023-02-24. Measure bound for translation surfaces with short saddle connections. https://doi.org/10.1007/s00039-023-00636-9

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