On the Spectral Gap of Negatively Curved Surface Covers

Given a negatively curved compact Riemannian surface X, we prove explicit estimates, valid with high probability as the degree goes to infinity, of the first non-trivial eigenvalue of the Laplacian on random Riemannian covers of X. The explicit gap depends on the bottom of the spectrum of the universal cover λ ₀. We first prove a lower bound that generalizes in variable curvature a result of Magee–Naud–Puder [24] known for hyperbolic surfaces. We then formulate a conjecture on the optimal spectral gap and show that if the sequence of representations associated to the covers converge strongly, the conjecture holds.