Zeta functions of geometrically finite graphs of groups

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.