A random cover of a compact hyperbolic surface has relative spectral gap $$\frac{3}{16}-\varepsilon $$ 3 16 - ε

Let X be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature \(-1\). For each \(n\in {\mathbf {N}}\), let \(X_{n}\) be a random degree-n cover of X sampled uniformly from all degree-n Riemannian covering spaces of X. An eigenvalue of X or \(X_{n}\) is an eigenvalue of the associated Laplacian operator \(\Delta _{X}\) or \(\Delta _{X_{n}}\). We say that an eigenvalue of \(X_{n}\) is new if it occurs with greater multiplicity than in X. We prove that for any \(\varepsilon >0\), with probability tending to 1 as \(n\rightarrow \infty \), there are no new eigenvalues of \(X_{n}\) below \(\frac{3}{16}-\varepsilon \). We conjecture that the same result holds with \(\frac{3}{16}\) replaced by \(\frac{1}{4}\).