Selberg, Ihara and Berkovich

TL;DR

This paper investigates the asymptotic behavior of resonances in degenerating Kleinian Schottky groups using zeta functions, connecting Archimedean and non-Archimedean properties, and extends results on Hausdorff dimension decay.

Contribution

It introduces an intermediate zeta function unifying Archimedean and non-Archimedean data, and proves convergence of Selberg zeta functions to Ihara zeta functions in degenerating families.

Findings

Convergence of rescaled Selberg zeta functions to Ihara zeta functions.

Exponential error term in Hausdorff dimension asymptotics.

Introduction of an intermediate zeta function capturing both properties.

Abstract

We use the Selberg zeta function to study the limit 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 to the relevant non-Archimedean Schottky group acting on the Berkovich projective line. Moreover, we show that these techniques can be used to get an exponential error term in a result of McMullen (recently extended by Dang and Mehmeti) about the asymptotics for the vanishing rate of the Hausdorff dimension of limit sets of certain degenerating Schottky groups generating symmetric three-funnel surfaces. Here, one key idea is to introduce an intermediate zeta function capturing \emph{both} non-Archimedean and Archimedean information (while the traditional Selberg, resp. Ihara zeta functions concern only Archimedean, resp. non-Archimedean properties).