Selberg's zeta function and the spectral geometry of geometrically finite hyperbolic surfaces

Abstract

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 $\mathrm{SL}(2,\mathbb{R})$ is determined by its trace spectrum up to finitely many possibilities, thus generalizing results of McKean [20] and Müller [23] to groups which are not necessarily cofinite.