Canonical measures on metric graphs and a Kazhdan's theorem
成果类型：
Article
署名作者：
Shokrieh, Farbod; Wu, Chenxi
署名单位：
Cornell University; Rutgers University System; Rutgers University New Brunswick
刊物名称：
INVENTIONES MATHEMATICAE
ISSN/ISSBN：
0020-9910
DOI：
10.1007/s00222-018-0838-5
发表日期：
2019
页码：
819-862
关键词：
摘要：
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.