Spectral gap of random covers of negatively curved noncompact surfaces

Let $(X,g)$ be a complete noncompact geometrically finite surface with pinched negative curvature $-b^2\le K_g\le -1$. Let ${\lambda }_0(\tilde{X})$ denote the bottom of the $L^2-$spectrum of the Laplacian on the universal cover $\tilde{X}$. We show that a uniformly random degree-$n$ cover $X_n$ of $X$ has no eigenvalues below ${\lambda }_0(\tilde{X})-\epsilon$ other than those of $X$ and with the same multiplicity, with probability tending to $1$ as $n\to \mathrm{\infty }$. This extends a result of Hide–Magee to metrics of pinched negative curvature.