Lifting di-analytic involutions of compact Klein surfaces to extended-Schottky uniformizations
Let $S$ be a compact Klein surface together with a di-analytic involution $\kappa :S \to S$. The lowest uniformizations of $S$ are those whose deck group is an extended-Schottky group, that is, an extended Kleinian group whose orientation preserving half is a Schottky group. If $S$ is a bordered compact Klein surface, then it is well known that $\kappa $ can be lifted with respect to a suitable extended-Schottky uniformization of $S$. In this paper, we complete the above lifting property by proving that if $S$ is a closed Klein surface, then $\kappa $ can also be lifted to a suitable extended-Schottky uniformization.