Work

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

Year: 2025

Type: article

Abstract: Cutting a hyperbolic surface $X$ X 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 $X$ X . This result strengthens Mirzakhani’s famous simple closed multi-geodesic counting theorems for hyperbolic surfaces.

Source: Inventiones mathematicae

DOI: 10.1007/s00222-025-01364-7

Authors: Francisco Arana–Herrera, Aaron Calderon

Institutions: University of Maryland, College Park, University of Chicago

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funder: National Science Foundation