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Limits of canonical forms on towers of Riemann surfaces

Research paper by Hyungryul Baik, Farbod Shokrieh, Chenxi Wu

Indexed on: 01 Aug '18Published on: 01 Aug '18Published in: arXiv - Mathematics - Algebraic Geometry



Abstract

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 $\{S_n \rightarrow S\}$ 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.