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. Zuk, 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 Gruryter, Berlin, 2004, pp. 141-180] 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. (c) 2008 Elsevier Inc. All rights reserved.

Guido, D., Isola, T., & Lapidus, M.L. (2008). Ihara's zeta function for periodic graphs and its approximation in the amenable case. JOURNAL OF FUNCTIONAL ANALYSIS, 255(6), 1339-1361 [10.1016/j.jfa.2008.07.011].

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

GUIDO, DANIELE;ISOLA, TOMMASO;
2008

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. Zuk, 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 Gruryter, Berlin, 2004, pp. 141-180] 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. (c) 2008 Elsevier Inc. All rights reserved.
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/05 - Analisi Matematica
English
Con Impact Factor ISI
Amenable graphs; Amenable groups; Analytic determinant; Approximation by finite graphs; Determinant formula; Functional equations; Ihara zeta function; Periodic graphs
Guido, D., Isola, T., & Lapidus, M.L. (2008). Ihara's zeta function for periodic graphs and its approximation in the amenable case. JOURNAL OF FUNCTIONAL ANALYSIS, 255(6), 1339-1361 [10.1016/j.jfa.2008.07.011].
Guido, D; Isola, T; Lapidus, M
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/27070
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