On the minimal diameter of closed hyperbolic surfaces

TL;DR

This paper establishes that the minimal diameter of large genus hyperbolic surfaces grows logarithmically with genus, using a probabilistic approach involving lattice point counting and random graph analysis.

Contribution

It provides the first asymptotic characterization of minimal diameter growth in hyperbolic surfaces of increasing genus.

Findings

Minimal diameter grows asymptotically as log g

Uses random constructions and lattice point counting

Analyzes random trivalent graphs for geometric insights

Abstract

We prove that the minimal diameter of a hyperbolic compact orientable surface of genus $g$ is asymptotic to $\log g$ as $g \to \infty$. The proof relies on a random construction, which we analyse using lattice point counting theory and the exploration of random trivalent graphs.