Spectral gap of random covers of negatively curved noncompact surfaces

2025/05/11 by Moy, Julien · 3 citations
#35P15 #58J35 #60B20 #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2505.07056

Abstract

Let (X,g) be a complete noncompact geometrically finite surface with pinched negative curvature -b²≤ K_g ≤ -1. Let λ₀(\widetildeX) denote the bottom of the L²-spectrum of the Laplacian on the universal cover \widetildeX. We show that a uniformly random degree-n cover X_n of X has no eigenvalues below λ₀(\widetildeX)-ε other than those of X and with the same multiplicity, with probability tending to 1 as n→ ∞. This extends a result of Hide--Magee to metrics of pinched negative curvature.

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