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We study resonances of surfaces of revolution obtained by removing a diskfrom a cone and attaching a hyperbolic cusp in its place. These surfacesinclude ones with nontrapping geodesic flow (every maximally extendednon-reflected geodesic is unbounded) and yet infinitely many long livingresonances (resonances with uniformly bounded imaginary part, i.e. decay rate).

Nontrapping surfaces of revolution with long living resonances