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

Connection link: https://hu-berlin.zoom.us/j/61686623112
Contact: Gaetan Borot (gaetan.borot@hu-berlin.de)

Cutting a hyperbolic surface along a simple closed multi-geodesic yields 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 hyperbolic surface. This result strengthens Mirzakhani’s famous simple close multi-geodesic counting theorems for hyperbolic surfaces. This is joint work with Aaron Calderon.