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 is determined by its trace spectrum up to finitely many possibilities, thus generalizing results of McKean  and Müller  to groups which are not necessarily cofinite.
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David Borthwick, Chris Judge, Peter A. Perry, Selberg's zeta function and the spectral geometry of geometrically finite hyperbolic surfaces. Comment. Math. Helv. 80 (2005), no. 3, pp. 483–515DOI 10.4171/CMH/23