  For hyperbolic Riemann surfaces of finite geometry, we study Selberg's zeta
function and its relation to the relative scattering phase and the resonances
of the Laplacian. As an application we show that the conjugacy class of a
finitely generated, torsion-free, discrete subgroup of SL(2,R) is determined by
its trace spectrum up to finitely many possibilities, thus generalizing results
of McKean and Mueller to groups which are not necessarily cofinite.

Differential Geometry;Spectral Theory