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

2025/01/07 by Lewis Bowen, Bowen, Lewis, Kasra Rafi +3
Biochemistry, Genetics and Molecular Biology · Computer Science · Social Sciences · #32G15 30F60 57M50 #FOS: Mathematics #Geometric Topology (math.GT) #Natural Language Processing Techniques #Probability (math.PR) #RNA and protein synthesis mechanisms #Wikis in Education and Collaboration

paper · pdf · doi:10.48550/arxiv.2501.03474

openalex publication_date 2025/01/07 · openalex created_date 2025/01/09 · openalex updated_date 2026/07/28

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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