Random covers of surfaces

TL;DR

This paper investigates the properties of random covers of hyperbolic surfaces with free fundamental groups, revealing their distribution on moduli space and calculating asymptotic geometric invariants like the systole.

Contribution

It introduces a new probabilistic model for random covers of hyperbolic surfaces with free fundamental groups and analyzes their distribution and geometric properties.

Findings

Distribution of random covers follows a specific measure on moduli space

Explicit calculations for the case k=2

Asymptotic expectations for systole and other invariants

Abstract

We study random covers of a closed hyperbolic surface $\Sigma$, subject to the condition that, for $k\geq 2$, the fundamental group is isomorphic to the free group $F_k$. We show that asymptotically they distribute according to a specific probability measure on the moduli space of metric graphs. As we will demonstrate with explicit calculations for $k=2$, this allows us to determine asymptotic values for the expectation of the systole and other geometric invariants of the covers.