Benjamini-Schramm limits of high genus translation surfaces: research announcement
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915092968243200 |
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| author | Bowen, Lewis Rafi, Kasra Vallejos, Hunter |
| author_facet | Bowen, Lewis Rafi, Kasra Vallejos, Hunter |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03474 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Benjamini-Schramm limits of high genus translation surfaces: research announcement Bowen, Lewis Rafi, Kasra Vallejos, Hunter Geometric Topology Probability 32G15 30F60 57M50 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. |
| title | Benjamini-Schramm limits of high genus translation surfaces: research announcement |
| topic | Geometric Topology Probability 32G15 30F60 57M50 |
| url | https://arxiv.org/abs/2501.03474 |