Types of complete infinitely sheeted planes

Abstract

We will answer negatively to the question whether the completeness of infinitely sheeted covering surfaces of the extended complex plane have anything to do with their types being parabolic or hyperbolic. This will be accomplished by giving a one parameter family $\{ W[\alpha] : \alpha \in {\cal A} \}$ of complete infinitely sheeted planes $W[\alpha]$ depending on the parameter set ${\cal A}$ of sequences $\alpha = (a_{n})_{n \geq 1}$ of real numbers $0 < a_{n} \leq 1/2$ $(n \geq 1)$ such that $W[\alpha]$ is parabolic for 'small' $\alpha$'s and hyperbolic for 'large' $\alpha$'s.