Benjamini-Schramm limits of high genus translation surfaces: research announcement

Lewis Bowen; Kasra Rafi; Hunter Vallejos

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arXiv · 2501.03474

Benjamini-Schramm limits of high genus translation surfaces: research announcement

Lewis Bowen·Kasra Rafi·Hunter Vallejos

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Abstract

We prove that the sequence of Masur-Smillie-Veech (MSV) distributed random translation surfaces, with area equal to genus, Benjamini-Schramm converges as genus tends to infinity. This means that for any fixed radius $r>0$, if $X_g$ is an MSV-distributed random translation surface with area $g$ and genus $g$, and $o$ is a uniformly random point in $X_g$, then the radius-$r$ neighborhood of $o$ in $X_g$, as a pointed measured metric space, converges in distribution to the radius $r$ neighborhood of the root in a Poisson translation plane, which is a random pointed surface we introduce here. Along the way, we obtain bounds on statistical local geometric properties of translation surfaces, such as the probability that the random point $o$ has injectivity radius at most $r$, which may be of independent interest.

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Lewis Bowen, Kasra Rafi, Hunter Vallejos. 2025-01-07. Benjamini-Schramm limits of high genus translation surfaces: research announcement. https://arxiv.org/abs/2501.03474

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