Zeta functions of geometrically finite graphs of groups

TL;DR

This paper studies zeta functions of infinite graphs of groups from cuspidal tree-lattices, revealing examples of non-isomorphic structures sharing zeta functions and analyzing spectral properties related to geodesic counting.

Contribution

It introduces new examples of non-isomorphic graphs with identical zeta functions and examines spectral convergence properties of these graphs.

Findings

Non-isomorphic graphs can have the same zeta function.

Spectral analysis shows pole-free regions approaching zero.

Small exponential error-term in geodesic counting formula.

Abstract

In this paper, we explore the properties of zeta functions associated with infinite graphs of groups that arise as quotients of cuspidal tree-lattices, including all non-uniform arithmetic quotients of the tree of rank one Lie groups over local fields. Through various examples, we illustrate pairs of non-isomorphic cuspidal tree-lattices with the same Ihara zeta function. Additionally, we analyze the spectral behavior of a sequence of graphs of groups whose pole-free regions of zeta functions converge towards 0, which also presents an example of arbitrary small exponential error-term in counting geodesic formula.