The shapes of complementary subsurfaces to simple closed hyperbolic multi-geodesics

• 成果类型: Article
• 署名作者: Arana-Herrera, Francisco; Calderon, Aaron
• 署名单位: University System of Maryland; University of Maryland College Park; University of Chicago
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01364-7
• 发表日期: 2025
• 页码: 571-626
• 关键词: moduli space; intersection theory; riemann surfaces; integer points; geometry; spheres; CURVES; number

摘要

Cutting a hyperbolic surface X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X$\end{document} along a simple closed multi-geodesic results in a hyperbolic structure on the complementary subsurface. We study the distribution of the shapes of these subsurfaces in moduli space as boundary lengths go to infinity, showing that they equidistribute to the Kontsevich measure on a corresponding moduli space of metric ribbon graphs. In particular, random subsurfaces look like random ribbon graphs, a law which does not depend on the initial choice of X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X$\end{document}. This result strengthens Mirzakhani's famous simple closed multi-geodesic counting theorems for hyperbolic surfaces.