Random covers of surfaces

2025/11/27 by Wright, Sophie
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2511.22731

Abstract

We study random covers of a closed hyperbolic surface Σ, subject to the condition that, for k≥ 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.

Citations

• Counting subgroups via Mirzakhani's curve counting
• Counting geodesics of given commutator length
• Counting Mapping Class group orbits on hyperbolic surfaces
• Injectivity Radius and Fundamental Group of Hyperbolic 3-Manifolds

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