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Transactions of the American Mathematical Society

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The Ihara zeta function for infinite graphs


Authors: Daniel Lenz, Felix Pogorzelski and Marcel Schmidt
Journal: Trans. Amer. Math. Soc. 371 (2019), 5687-5729
MSC (2010): Primary 05C63; Secondary 05C38, 05C80, 22D40, 46L51
DOI: https://doi.org/10.1090/tran/7508
Published electronically: September 18, 2018
MathSciNet review: 3937307
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Abstract: We put forward the concept of measure graphs. These are (possibly uncountable) graphs equipped with an action of a groupoid and a measure invariant under this action. Examples include finite graphs, periodic graphs, graphings, and percolation graphs. Making use of Connes's noncommutative integration theory, we construct a zeta function and present a determinant formula for it. We further introduce a notion of weak convergence of measure graphs and show that our construction is compatible with it. The approximation of the Ihara zeta function via the normalized version on finite graphs in the sense of Benjamini and Schramm follows as a special case. Our framework not only unifies corresponding earlier results occurring in the literature. It likewise provides extensions to rich new classes of objects such as percolation graphs.


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Additional Information

Daniel Lenz
Affiliation: Mathematisches Institut, Friedrich Schiller Universität Jena, 07743 Jena, Germany
Email: daniel.lenz@uni-jena.de

Felix Pogorzelski
Affiliation: Mathematisches Institut, Universität Leipzig, 04109 Leipzig, Germany
Email: felix.pogorzelski@math.uni-leipzig.de

Marcel Schmidt
Affiliation: Mathematisches Institut, Friedrich Schiller Universität Jena, 07743 Jena, Germany
Email: schmidt.marcel@uni-jena.de

DOI: https://doi.org/10.1090/tran/7508
Received by editor(s): February 20, 2017
Received by editor(s) in revised form: November 7, 2017, December 16, 2017, and January 8, 2018
Published electronically: September 18, 2018
Additional Notes: The second author expresses his thanks for support through the German National Academic Foundation (Studienstiftung des deutschen Volkes).
The third author has been financially supported by the Graduiertenkolleg 1523/2: Quantum and gravitational fields
Article copyright: © Copyright 2018 American Mathematical Society