Low-lying resonances for infinite-area hyperbolic surfaces with long closed geodesics
Abstract
We consider sequences (Xn)n∈ N of coverings of convex cocompact hyperbolic surfaces X with Euler characterictic (Xn) tending to -∞ as n ∞. We prove that for n large enough, each Xn has an abundance of "low-lying" resonances, provided the length of the shortest closed geodesic on Xn grows sufficiently fast. When applied to congruence covers we obtain a bound that improves upon a result of Jakobson, Naud, and the author in JNS. Our proof uses the wave 0-trace formula of Guillop\'e--Zworski GZ99 together with specifically tailored test-functions with rapidly decaying Fourier transform.
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