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

TL;DR

This paper investigates how the shapes of subsurfaces obtained by cutting hyperbolic surfaces along multi-geodesics distribute in moduli space, revealing they become equidistributed to the Kontsevich measure as boundary lengths grow large.

Contribution

It establishes the equidistribution of subsurface shapes to the Kontsevich measure, connecting hyperbolic geometry with ribbon graph models and extending Mirzakhani's counting results.

Findings

Subsurface shapes become equidistributed in moduli space as boundary lengths increase.

Random subsurfaces resemble random ribbon graphs, independent of initial surface.

Results strengthen the connection between hyperbolic geometry and combinatorial models.

Abstract

Cutting a hyperbolic surface 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. This result strengthens Mirzakhani's famous simple closed multi-geodesic counting theorems for hyperbolic surfaces.