Automorphic functions on general domains in \(\mathbb{C}^ n\) (Q1922126)

Let \(U\subset \mathbb{C}^n\) be connected, and suppose \(\Gamma\) acts properly discontinuously biholomorphically on \(U\). Let \(K(\Gamma,U)\) denote the field of \(\Gamma\)-automorphic functions on \(U\). If there is some \(\delta>0\) such that \(\int_U (1 + |z |^2 )^\delta < \infty\), then \(K(\Gamma,U)\) has transcendence degree at least \(n\) and separates points of \(\Gamma \backslash U\). If \(\Gamma \backslash U\) is not compact, then the transcendence degree of \(K(\Gamma,U)\) has relatively little to say about the analysis or geometry of \(\Gamma \backslash U\), as is demonstrated by the corresponding counterexample.