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

Understand

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 $X$.