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We study the isoresonance problem on non-compact surfaces of finite area thatare hyperbolic outside a compact set. Inverse resonance problems correspond toinverse spectral problems in the non-compact setting. We consider a conformalclass of surfaces with hyperbolic cusps where the deformation takes placeinside a fixed compact set. Inside this compactly supported conformal class weconsider isoresonant metrics, i.e. metrics for which the set of resonances isthe same, including multiplicities. We prove that sets of isoresonant metricsinside the conformal class are sequentially compact. We use relativedeterminants, splitting formulae for determinants and the result of B. Osgood,R. Phillips and P. Sarnak about compactness of sets of isospectral metrics onclosed surfaces.  In the second part, we study the relative determinant of the Laplace operatoron a hyperbolic surface as function on the moduli space. We consider the modulispace of hyperbolic surfaces of fixed genus and fixed number of cusps. Weconsider the relative determinant of the Laplace operator and a model operatordefined on the cusps. We prove that the relative determinant tends to zero asone approaches the boundary of the moduli space.

Isoresonant conformal surfaces with cusps and boundedness of the relative determinant