Ihara's zeta function for periodic graphs and its approximation in the amenable case |
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Authors: | Daniele Guido Tommaso Isola |
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Affiliation: | a Dipartimento di Matematica, Università di Roma “Tor Vergata”, I-00133 Roma, Italy b Department of Mathematics, University of California, Riverside, CA 92521-0135, USA |
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Abstract: | In this paper, we give a more direct proof of the results by Clair and Mokhtari-Sharghi [B. Clair, S. Mokhtari-Sharghi, Zeta functions of discrete groups acting on trees, J. Algebra 237 (2001) 591-620] 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 in [R.I. Grigorchuk, A. ?uk, The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps, in: V.A. Kaimanovich, et al. (Eds.), Proc. Workshop, Random Walks and Geometry, Vienna, 2001, de Gruyter, Berlin, 2004, pp. 141-180] by Grigorchuk and ?uk. 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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Keywords: | Periodic graphs Ihara zeta function Analytic determinant Determinant formula Functional equations Amenable groups Amenable graphs Approximation by finite graphs |
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