A trace on fractal graphs and the Ihara zeta function

Abstract

Abstract. Starting with Ihara's work in 1968, there has been a growing interest in the study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and Terras, Mizuno and Sato, to name just a few authors. Then, Clair and Mokhtari-Sharghi studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms. The main formula in all these treatments establishes a connection between the zeta function, originally defined as an infinite product, and the Laplacian of the graph. In this article, we consider a different class of infinite graphs. They are fractal graphs, i.e. they enjoy a selfsimilarity property. We define a zeta function for these graphs and, using the machinery of operator algebras, we prove a determinant formula, which relates the zeta function with the Laplacian of the graph. We also prove functional equations, and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs. IntroductionThe Ihara zeta function, originally associated to certain groups and then combinatorially reinterpreted as associated with finite graphs or with their infinite coverings, is defined here for a new class of infinite graphs, called self-similar graphs. The corresponding determinant formula and functional equations are established.The combinatorial nature of the Ihara zeta function was first observed by Serre (see [37], Introduction), but it was only through the works of Sunada [42], Hashimoto [19,20] and Bass [4] that it became a graph-theoretical object, at the same time keeping some number-theoretically flavoured properties, such as the Euler product formula or the functional equation.The Ihara zeta function [24] was written as an infinite product (Euler product) over G-conjugacy classes of primitive elements in a group G, namely elements whose centralizer (in G) is generated by the element itself. As explained in detail in the Key words and phrases. Self-similar fractal graphs, Ihara zeta function, geometric operators, C*-algebra, analytic determinant, determinant formula, primitive cycles, Euler product, functional equations, amenable graphs, approximation by finite graphs.The first and second authors were partially supported by MIUR, GNAMPA and by the European Network "Quantum Spaces -Noncommutative Geometry" HPRN-CT-2002-00280.The third author was partially supported by the National Science Foundation, the Academic Senate of the University of California, and GNAMPA. introductions of [4,39], Ihara's construction can be rephrased in terms of a regular (i.e. constant number of edges spreading from each vertex) finite graph X, its universal covering Y and the corresponding structure group G = π 1 (X). By the homotopic nature of G, one may equivalently represent G-conjugacy classes in terms of suitably reduced primitive cycles on the graph X. Here, a reduced cycle on X (of length m) is a set {e j , j ∈ Z m }, where the starting vertex of e j+1 coincides with the ending vertex of e j , and e j+1 is not the opposite of e j , for j ∈ Z m . Beside...

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