Convergence of Zeta Functions of Graphs

The L-zeta function of an infinite graph Y (defined previously in a ball around zero) has an analytic extension. For a tower of finite graphs covered by Y , the normalized zeta functions of the finite graphs converge to the L-zeta function of Y . Introduction Associated to any finite graph X there is a zeta function Z(X,u), u ∈ C. It is defined as an infinite product but shown (in various different cases) by Ihara, Hashimoto, and Bass [5, 4, 1] to be a polynomial. Indeed the rationality formula for a q + 1 regular X states that: Z(X,u) = (1− u2)−χ(X) Det(I − δu+ qu). (0.1) Here δ is the adjacency operator of X . In [2], an L-zeta function is defined for noncompact graphs with symmetries, using the machinery of von Neumann algebras. A rationality formula similar to (0.1) expresses the relationship between the zeta function and the von Neumann determinant of a Laplace operator. The results of this paper focus on an especial case. Let Y be an infinite graph which covers a finite graph B = π\Y . The L-zeta function Zπ(Y, u) is defined in [2] only in a small neighborhood of zero. The first result of this paper is to extend the L-zeta function to the interior of C = {u ∈ C : |u| = q−1/2} ∪ [−1,− 1 q ] ∪ [ 1 q , 1]. In the second part of the paper, we consider a tower of finite graphs Bi covered by Y . Put Ni = |Bi|/|B|. In Theorem 2.1 we show that the zeta functions for the Bi, renormalized by taking N th i roots, converge to the L -zeta function for Y . The argument is inspired by, and uses, work of Lück [7]. Date: July 14, 2000. 1 2 BRYAN CLAIR AND SHAHRIAR MOKHTARI-SHARGHI In the first section we recall the definitions of the zeta functions of finite and infinite graphs. One of the main results of this paper is Theorem 1.5 in this section. In the second section of this paper we prove the convergence theorem and exhibit interesting examples. Theorem 2.4 generalizes work of Deitmar [3]. 1. Zeta functions In this section we recall the definition of the zeta function and related material. We first recall the definition of the zeta function for finite graphs. 1.1. Finite Graphs. For a graph X , let V X and EX denote the sets of vertices and edges, respectively, of X . If each vertex has the same degree then X is regular. Definition 1.1. Let X be a finite graph. A closed path in X is primitive if it is not a nontrivial power of another path inside the fundamental group of X . Let P be the set of free homoptopy classes of primitive closed paths of X . Then the zeta function of X is