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Canonical measures on metric graphs and a Kazhdan’s theorem

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Abstract

We extend the notion of canonical measures to all (possibly non-compact) metric graphs. This will allow us to introduce a notion of “hyperbolic measures” on universal covers of metric graphs. Kazhdan’s theorem for Riemann surfaces describes the limiting behavior of canonical (Arakelov) measures on finite covers in relation to the hyperbolic measure. We will prove a generalized version of this theorem for metric graphs, allowing any infinite Galois cover to replace the universal cover. We will show all such limiting measures satisfy a version of Gauss–Bonnet formula which, using the theory of von Neumann dimensions, can be interpreted as a “trace formula”. In the special case where the infinite cover is the universal cover, we will provide explicit methods to compute the corresponding limiting (hyperbolic) measure. Our ideas are motivated by non-Archimedean analytic and tropical geometry.

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Acknowledgements

We would like to thank Matthew Baker and Curtis McMullen for helpful remarks and suggestions. We also thank Omid Amini, Matthew Baker, Robin de Jong, and the anonymous referee for valuable comments on an earlier draft.

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Authors and Affiliations

  1. Cornell University, Ithaca, NY, 14853-4201, USA

    Farbod Shokrieh

  2. Rutgers University, Piscataway, NJ, 08854-8019, USA

    Chenxi Wu

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  1. Farbod Shokrieh
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Correspondence to Farbod Shokrieh.

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Shokrieh, F., Wu, C. Canonical measures on metric graphs and a Kazhdan’s theorem. Invent. math. 215, 819–862 (2019). https://doi.org/10.1007/s00222-018-0838-5

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