Canonical measures on metric graphs and a Kazhdan's theorem

TL;DR

This paper generalizes canonical measures to all metric graphs, introduces hyperbolic measures on their covers, and proves a Kazhdan's theorem analogue, linking limiting measures to a trace formula with applications in non-Archimedean and tropical geometry.

Contribution

It extends canonical measures to non-compact metric graphs and proves a Kazhdan's theorem analogue for infinite Galois covers, with explicit computations in special cases.

Findings

Generalized Kazhdan's theorem for metric graphs

Limiting measures satisfy a Gauss-Bonnet type formula

Explicit methods for hyperbolic measure computation in universal covers

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.