  This paper has been withdrawn because of a mistake in Lemma 7.4. For a
corrected version of the main theorem, as well as various further developments
and applications, see arXiv:1210.7736. For nonpositively curved perturbations
of parabolic cylinders we establish the existence of a logarithmically large
resonance free region. We use an escape function construction in a compact part
of the manifold and near infinity we use the method of complex scaling. To the
author's knowledge this is the first proof of a large resonance free region for
manifolds with cusps.
