  We introduce a notion of relative isospectrality for surfaces with boundary
having possibly non-compact ends either conformally compact or asymptotic to
cusps. We obtain a compactness result for such families via a conformal surgery
that allows us to reduce to the case of surfaces hyperbolic near infinity
recently studied by Borthwick and Perry, or to the closed case by Osgood,
Phillips and Sarnak if there are only cusps.
