Large genus bounds for the distribution of triangulated surfaces in moduli space

Abstract

Triangulated surfaces are Riemann surfaces which admit a conformal triangulation by equilateral triangles. We show that in the large genus setting, triangulated surfaces are well-distributed in moduli space. This was first predicted by Brooks and Makover in 2004, and since then many more results have provided evidence that the large genus geometry of random triangulated surfaces mirrors the large genus geometry of random hyperbolic Riemann surfaces.