Ihara's zeta function for periodic graphs and its approximation in the amenable case

Details Ihara's zeta function for periodic graphs and its approximation in the amenable case Ihara's zeta function for periodic graphs and its approximation in the amenable case

2006-08-09

arxiv.org/abs/math/0608229v1

Journal of Functional Analysis 255 (2008) no. 6, 1339-1361

Mathematics

Operator Algebras

21 pages, 4 figures

Scientific paper

10.1016/j.jfa.2008.07.011

In this paper, we give a more direct proof of the results by Clair and Mokhtari-Sharghi on the zeta functions of periodic graphs. In particular, using appropriate operator-algebraic techniques, we establish a determinant formula in this context and examine its consequences for the Ihara zeta function. Moreover, we answer in the affirmative one of the questions raised by Grigorchuk and Zuk. Accordingly, we show that the zeta function of a periodic graph with an amenable group action is the limit of the zeta functions of a suitable sequence of finite subgraphs.

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