Alon and Bilu–Linial conjectures for random and two-covers of hyperbolic surfaces

Let $\Gamma$ be a free group and let $X=\Gamma\backslash\mathbb{H}$. For an onto homomorphism $\phi:\Gamma\to S_n$, let $X'$ be the corresponding index-$n$ cover of $X$. A new eigenvalue is an eigenvalue of the cover not inherited from the base surface.

Alon and Bilu–Linial conjectures. (1) For every $\epsilon>0$, with probability $1-o(1)$ as $n\to\infty$, every new eigenvalue $\lambda$ of an $n$-cover $X'$ satisfies

$$\lambda\geq \frac{1}{4}+\epsilon.$$

(2) There exists a $2$-cover $X'$ of $X$ such that every new eigenvalue $\lambda$ of $X'$ satisfies

$$\lambda\geq \frac{1}{4}.$$

These are continuous analogues of well-known conjectures for graphs: the first is Alon's conjecture, proved for graphs by Friedman, while the second is Bilu–Linial's conjecture, solved in the bipartite graph case. Their status for the hyperbolic-surface covers considered here is left open.