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We consider sequences $(X_n)_{n\in \mathbb{N}}$ of coverings of convexcocompact hyperbolic surfaces $X$ with Euler characterictic $\chi(X_n)$ tendingto $-\infty$ as $n\to \infty.$ We prove that for $n$ large enough, each $X_n$has an abundance of "low-lying" resonances, provided the length of the shortestclosed geodesic on $X_n$ grows sufficiently fast. When applied to congruencecovers we obtain a bound that improves upon a result of Jakobson, Naud, and theauthor in \cite{JNS}. Our proof uses the wave 0-trace formula ofGuillop\'{e}--Zworski \cite{GZ99} together with specifically tailoredtest-functions with rapidly decaying Fourier transform.

Low-lying resonances for infinite-area hyperbolic surfaces with long closed geodesics