For a certain class of smooth bounded domains in \(\mathbb C\), let \(\tilde\Delta\) denote the Laplace--Beltrami operator for the metric that pulls back to the hyperbolic metric on the universal covering space, the unit disc. It is shown that the Green function of \(\tilde{\Delta}^2\) is positive. For simply connected domains, this follows from the conformal invariance of \(\tilde{\Delta}\) and an explicit formula in the case of the disc. For multiply connected domains convergence estimates for sums over the deck transformation group \(G\) enter. In similar spirit, a Plancherel formula for \(\tilde{\Delta}\) is derived. These calculations are made explicit for the annulus, where they involve the hypergeometric function \({}_2F_1\). Finally examples of domains with \(k\geq2\) holes are constructed such that the abscissa of convergence \(s_0\) of the series \(\sum_{\omega\in G}(1-|\omega(0)|^2)^s\) is arbitrarily small \(>0\), or else is arbitrarily little below \(\log(\sqrt{3})/\log(1+\sqrt{2})\approx 0.62\). The hypothesis on the domain needs \(s_0\leq1/2\) and follows from \(s_0<1/2\).