Spectral gap of random covers of negatively curved noncompact surfaces

Understand

Let $(X,g)$ be a complete noncompact geometrically finite surface with pinched negative curvature $-b^2\leq K_g \leq -1$.

• Let $\lambda_0(\widetilde{X})$ denote the bottom of the $L^2-$spectrum of the Laplacian on the universal cover $\widetilde{X}$.
• We show that a uniformly random degree-$n$ cover $X_n$ of $X$ has no eigenvalues below $\lambda_0(\widetilde{X})-\varepsilon$ other than those of $X$ and with the same multiplicity, with probability tending to $1$ as $n\to \infty$.
• This extends a result of Hide--Magee to metrics of pinched negative curvature.