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

By David Borthwick, C. Judge, Peter Perry. 2003

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

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f1a32f89ff5c9822dd734a9fee7428e550e542b6