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Counting Cylinders on Z-covers of Genus 2 Square-tiled Surfaces
Abstract
We count maximal cylinders on zero holonomy ℤ-covers of genus $2$ square-tiled surfaces, up to ℤ-action, obtaining quadratic asymptotics. We also show that the leading term of the asymptotic, called the Siegel-Veech constant, can be recovered via a large-genus approximation by intermediate finite covers. Our work applies to the infinite staircases introduced by P. Hubert and G. Weitze-Schmithüsen. For many members of this family, we explicitly compute the associated Siegel-Veech constants. In particular, we exhibit the first infinite family of examples of zero holonomy ℤ-cover in which the number of cylinders grows sub-quadratically.
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BibTeX
@misc{artigiani2026counting,
title = {{Counting Cylinders on Z-covers of Genus 2 Square-tiled Surfaces}},
author = {Mauro Artigiani and Angel Pardo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06609},
eprint = {2610.06609},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06609},
}
RIS
TY - UNPB
TI - Counting Cylinders on Z-covers of Genus 2 Square-tiled Surfaces
AU - Mauro Artigiani
AU - Angel Pardo
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06609
UR - https://arxiv.org/abs/2610.06609
AB - We count maximal cylinders on zero holonomy ℤ-covers of genus $2$ square-tiled surfaces, up to ℤ-action, obtaining quadratic asymptotics. We also show that the leading term of the asymptotic, called the Siegel-Veech constant, can be recovered via a large-genus approximation by intermediate finite covers. Our work applies to the infinite staircases introduced by P. Hubert and G. Weitze-Schmithüsen. For many members of this family, we explicitly compute the associated Siegel-Veech constants. In particular, we exhibit the first infinite family of examples of zero holonomy ℤ-cover in which the number of cylinders grows sub-quadratically.
ER -