The role of compactification theory in the type problem

Abstract

We are concerned with the type problem for the covering surface $S_{\Gamma}$ or more precisely $(S_{\Gamma},S,\pi_{\Gamma})$ of the base parabolic Riemann surface $S$ with its projection $\pi_{\Gamma}$, where $S_{\Gamma}$ is the infinitely sheeted covering Riemann surface constructed from the sequence of replicas $S_{n}$ of $S$ and the family $\Gamma=\{\gamma_{n}:n \in \Bbb N \}$ of pasting arcs $\gamma_{n} \subset S$ with $\gamma_{n-1} \cap \gamma_{n}=\emptyset\(n \in \Bbb N)$ in such a fashion that $S_{n}\setminus(\gamma_{n-1}\cup\gamma_{n})$ is joined to $S_{n+1}\setminus(\gamma_{n}\cup\gamma_{n+1})$ crosswise along $\gamma_{n}$ for each $n \in \Bbb N$. Here $\Bbb N$ is the set of positive integers and the parabolicity of a surface is characterized by the nonexistence of the Green function on the surface. The central object of this paper is to show by using the theory of Wiener and Royden compactifications that the type of the covering surface $S_{\Gamma}$ is parabolic if the sequence of capacities ${\rm cap}(\gamma_{n},S\setminus\gamma_{0})$ of $\gamma_{n}\in\Gamma$ with respect to the surface $S$ less the arc $\gamma_{0}$ fixed in $S$ disjoint from all other arcs $\gamma_{n}\in\Gamma$ for all sufficiently large $n\in\Bbb N$ converges to zero so rapidly as to satisfy the condition $\sum_{n\in\Bbb N}{\rm cap} (\gamma_{n}, S\setminus\gamma_{0})<+\infty$.