Limits of canonical forms on towers of Riemann surfaces

TL;DR

This paper generalizes Kazhdan's theorem for canonical forms on Riemann surfaces, showing convergence of forms on infinite Galois covers and establishing a Gauss-Bonnet type theorem in this broader context.

Contribution

It extends Kazhdan's theorem to arbitrary infinite Galois covers and proves a new Gauss-Bonnet type theorem for such covers.

Findings

Canonical forms on finite covers converge uniformly to the hyperbolic form.

Generalization of Kazhdan's theorem to infinite Galois covers.

A Gauss-Bonnet type theorem for infinite Galois covers.

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.