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Spectral gaps without the pressure condition. (English) Zbl 1392.37065
The main theorem of this paper states that every convex co-compact hyperbolic surface has an essential spectral gap. The latter means that there is a strip beyond the unitarity axis in which the Selberg zeta function has only finitely many zeroes. Contrary to the previous results, the pressure condition is not required. The main new tool is the fractal uncertainty principle.

##### MSC:
 37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry 37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems 37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. 30D60 Quasi-analytic and other classes of functions of one complex variable 35B34 Resonance in context of PDEs 35P05 General topics in linear spectral theory for PDEs 35R01 PDEs on manifolds
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