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

2023/05/15 by Soares, Louis
#58J50 (Primary) 11F72 (Secondary) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2305.08713

Abstract

We consider sequences (X_n)_{n∈ ℕ} of coverings of convex cocompact hyperbolic surfaces X with Euler characterictic χ(X_n) tending to -∞ as n→ ∞. 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 \citeJNS. Our proof uses the wave 0-trace formula of Guillopé--Zworski \citeGZ99 together with specifically tailored test-functions with rapidly decaying Fourier transform.

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