  We use semiclassical propagation of singularities to give a general method
for gluing together resolvent estimates. As an application we prove estimates
for the analytic continuation of the resolvent of a Schr\"odinger operator for
certain asymptotically hyperbolic manifolds in the presence of trapping which
is sufficiently mild in one of several senses. As a corollary we obtain local
exponential decay for the wave propagator and local smoothing for the
Schr\"odinger propagator.
