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. (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;
2008-01-01
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.