通用平移面循环盖的西格尔-维奇常数

arXiv - MATH - Geometric Topology Pub Date : 2024-09-10 DOI:arxiv-2409.06600

David Aulicino, Aaron Calderon, Carlos Matheus, Nick Salter, Martin Schmoll

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引用次数: 0

摘要

我们计算了任意层中任意一般平移面的任意环状分支盖上圆柱体的渐近数量，这些圆柱体按其面积加权到任意非负幂。我们的公式只依赖于盖的拓扑不变式和度的数论性质：特别是，盖的位点和底层分量的相关西格尔-维奇常数的比值与分支值的数量无关。一个令人惊奇的推论是，对于 $area^3$ 西格尔-维奇常数来说，这个比率总是等于盖度的倒数。其中的一个关键要素是对某些环支盖的位置的连通成分的分类。

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Siegel-Veech Constants for Cyclic Covers of Generic Translation Surfaces

We compute the asymptotic number of cylinders, weighted by their area to any non-negative power, on any cyclic branched cover of any generic translation surface in any stratum. Our formulas depend only on topological invariants of the cover and number-theoretic properties of the degree: in particular, the ratio of the related Siegel-Veech constants for the locus of covers and for the base stratum component is independent of the number of branch values. One surprising corollary is that this ratio for $area^3$ Siegel-Veech constants is always equal to the reciprocal of the the degree of the cover. A key ingredient is a classification of the connected components of certain loci of cyclic branched covers.

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arXiv - MATH - Geometric Topology

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