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

We consider sequences $(X_n{)}_{n\in \mathbb{N}}$ of coverings of convex cocompact hyperbolic surfaces $X$ with Euler characterictic $\chi (X_n)$ tending to $-\mathrm{\infty }$ as $n\to \mathrm{\infty }\mathrm{.}$ We prove that for $n$ large enough, each $X_n$ has an abundance of “low-lying” resonances, provided the length of the shortest closed geodesic on $X_n$ grows sufficiently fast. When applied to congruence covers we obtain a bound that improves upon a result of Jakobson, Naud, and the author in \cite{JNS}. Our proof uses the wave 0-trace formula of Guillopé–Zworski \cite{GZ99} together with specifically tailored test-functions with rapidly decaying Fourier transform.