Selberg, Ihara and Berkovich

Abstract

We use the Selberg zeta function to analyze the limiting behavior of resonances in a degenerating family of Kleinian Schottky groups. We prove that, after a suitable rescaling, the Selberg zeta functions converge to the Ihara zeta function of a limiting finite graph associated with the relevant non-Archimedean Schottky group acting on the Berkovich projective line.

To reveal more interesting and intricate structures of resonances, we introduce intermediate zeta functions that encapsulate both Archimedean and non-Archimedean information – whereas the classical Selberg and Ihara zeta functions reflect only Archimedean and non-Archimedean properties, respectively. As an application of our techniques, we derive an exponential error term in a result of McMullen (recently extended by Dang–Mehmeti) concerning the asymptotic vanishing rate of the Hausdorff dimension of limit sets of certain degenerating Schottky groups generating symmetric three-funnel surfaces.