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 $\left\{S_n\to S\right\}$ 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.

中文翻译：

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黎曼曲面塔上规范形式的极限

我们证明了黎曼曲面上规范形式的Kazhdan定理的广义形式。在经典版本中，一个序列以升序开始$\left\{{\mathrm{小号}}_{ñ}\to \mathrm{小号}\right\}$双曲黎曼曲面S的有限Galois覆盖的一个集合，收敛到通用覆盖集合。定理指出，S上的形式序列是从S上的规范形式继承而来的${\mathrm{小号}}_{ñ}$一致地收敛到双曲形式（的倍数）。我们证明了该定理的广义形式，其中万有掩盖被任何无限的Galois掩盖代替。在此过程中，我们还证明了在任意无限Galois覆盖范围内的Gauss-Bonnet型定理。