  We construct a determinant of the Laplacian for infinite-area surfaces which
are hyperbolic near infinity and without cusps. In the case of a convex
co-compact hyperbolic metric, the determinant can be related to the Selberg
zeta function and thus shown to be an entire function of order two with zeros
at the eigenvalues and resonances of the Laplacian. In the hyperbolic near
infinity case the determinant is analyzed through the zeta-regularized relative
determinant for a conformal metric perturbation. We establish that this
relative determinant is a ratio of entire functions of order two with divisor
corresponding to eigenvalues and resonances of the perturbed and unperturbed
metrics. These results are applied to the problem of compactness in the smooth
topology for the class of metrics with a given set of eigenvalues and
resonances.
