Limits of canonical forms on towers of Riemann surfaces

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 $\{S_n\to S\}$ 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.