Abstract
We prove that any ergodic \(SL_2({\mathbb {R}})\)-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 \(O(\epsilon _1^2 \cdot \epsilon _2^2)\). 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.
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Notes
This is the definition we use at a point X in the smooth locus \({\mathcal {M}}^*\subset {\mathcal {M}}\), since at such a point \(T_X{\mathcal {M}}\) is a single linear subspace. At points of \({\mathcal {M}}-{\mathcal {M}}^*\), \(T_X{\mathcal {M}}\) is not a single linear subspace, but rather a finite union of linear subspaces, so the above definition needs modification. However \(\mu _{\mathcal {M}}({\mathcal {M}}-{\mathcal {M}}^*)=0\), so for the purposes of Theorem 1.3, it does not matter how we define \({\mathcal {M}}\)-independence at such X. For concreteness, one can define any set of saddle connections to be \({\mathcal {M}}\)-independent at \(X\in {\mathcal {M}}-{\mathcal {M}}^*\).
Theorem 1.3 also holds for strata in which the zeros are unlabeled. These strata are quotients of the labeled strata by an appropriate finite subgroup of the symmetric group, so they share the same qualitative properties.
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Acknowledgements
I am very grateful to Alex Wright for suggesting the problem and for helpful conversations and guidance. I also would like to thank Matt Bainbridge and Sam Grushevsky for patiently explaining their work to me. I thank Frederik Benirschke, Dawei Chen, Martin Möller, and Jenya Sapir for useful discussions. Finally, I gratefully acknowledge the support of the Fields Institute during the Thematic Program on Teichmüller Theory and its Connections to Geometry, Topology and Dynamics, where some of this work was done.
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Dozier, B. Measure bound for translation surfaces with short saddle connections. Geom. Funct. Anal. 33, 421–467 (2023). https://doi.org/10.1007/s00039-023-00636-9
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DOI: https://doi.org/10.1007/s00039-023-00636-9