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

2003/10/22 by D. Borthwick, David Borthwick, Christopher M. Judge +6
Mathematics · #58J50 #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Spectral Theory (math.SP) #math.DG #math.SP #msc:58J50

paper · pdf · doi:10.48550/arxiv.math/0310364

AMS-LaTeX, 27 pages, 3 figures. Revision adds references and corrects typos

openalex publication_date 2003/10/22 · arxiv created 2003/11/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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.

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