  Let $X=\Gamma\backslash\mathbb{H}^2$ be a convex cocompact hyperbolic
surface, let $X_n=\Gamma_n\backslash\mathbb{H}^2$ be an infinite family of
finite regular covers of $X$, and let $\mathcal{G}_n$ be the associated family
of Cayley graphs of the covering groups $G_n = \Gamma/\Gamma_n$ with respect to
the set of Schottky generators of $X$. Motivated by the work of Brooks
\cite{Brooks}, Burger \cite{Burger1, Burger2}, and Bourgain-Gamburd-Sarnak
\cite{BGS}, we conjecture that the surfaces $X_n$ have a uniform resonance-free
strip if the graphs $\mathcal{G}_n$ form a family of expanders. We prove some
results towards this conjecture. In particular, we confirm the conjecture under
two additional assumptions: the $\mathcal{G}_n$'s form a family of two-sided
expanders and the non-trivial irreducible representations of the groups $G_n$
are "large" compared to the size of $G_n$. Combining this result with the work
on expanders by Breuillard-Green-Guralnick-Tao \cite{BGGT}, we show that for
every finite simple group of Lie type $G$, "almost all" $G$-covers of $X$ have
a uniform resonance-free strip. Another consequence of our results is a new
proof of the result of Oh-Winter \cite{OhWinter} on uniform resonance-free
strips for congruence covers of convex cocompact hyperbolic surfaces. Our
proofs rely on representation theoretic and transfer operator techniques.
