Limits of canonical forms on towers of Riemann surfaces

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Hyungryul Baik, Farbod Shokrieh, Chenxi Wu

We prove a generalized version of Kazhdan's theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence { S n → S } of finite Galois covers of a hyperbolic Riemann surface S, converging to the universal cover. The theorem states that the sequence of forms on S inherited from the canonical forms on S n 's converges uniformly to (a multiple of) the hyperbolic form. We prove a generalized version of this theorem, where the universal cover is replaced with any infinite Galois cover. Along the way, we also prove a Gauss-Bonnet-type theorem in the context of arbitrary infinite Galois covers.

Original languageEnglish
JournalJournal fur die Reine und Angewandte Mathematik
Publication statusE-pub ahead of print - 2019

ID: 223822901