2017/09/01 by Louis Soares, Soares, Louis · 1 citation
Mathematics · #11J36 (Primary) #58J50 (Secondary) #Analytic Number Theory Research #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Spectral Theory (math.SP) #math.SP #msc:11J36 #msc:58J50
paper · pdf · doi:10.48550/arxiv.1709.00295
arxiv created 2017/09/01 · openalex publication_date 2017/09/01 · arxiv updated 2017/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X = Γ∖ ℍ be a non-elementary geometrically finite hyperbolic surface and let δ denote the Hausdorff dimension of the limit set Λ(Γ) . We prove that for every ε > 0 the surface X admits a finite cover X' such that the Selberg zeta function associated to X' has a zero s≠ δ with | δ- s| < ε . For δ> (1)/(2) we exploit the combinatorial interpretation of spectral gap in terms of expander graphs. For δ≤ (1)/(2) the proof is based on the thermodynamic formalism approach for L-functions associated to hyperbolic surfaces and an analogue of the Artin-Takagi formula for these L-functions.