Lowest uniformizations of compact Klein surfaces (Q1941179)

This paper treats discontinuous subgroups of the group of extended Möbius transformations \(\hat{\mathbb{M}}\). In the paper, a uniformization \((\Omega, G, P:\Omega\to S)\) of a Klein surface \(S\) means the one by an extended function group \(G\). Thus \(G\) is a discontinuous subgroup of \(\hat{\mathbb{M}}\) with an invariant component \(\Omega\) on which the orientation preserving half \(G^+\) of \(G\) acts freely, and \(P\) is the projection onto \(S\) identified with \(\Omega/G\). It suffices to consider only such uniformizations because any uniformization by a planer Riemann surface is equivalent to the one realized in this way (see, for example, X.F.16 in \textit{B. Maskit}'s [Kleinian groups. Berlin etc.: Springer (1988; Zbl 0627.30039)]). Let \(S\) be a compact Klein surface. The set of uniformizations of \(S\) is partially ordered: A uniformization \((\Omega_1, G_1, P_1:\Omega_1\to S)\) of \(S\) is higher than another uniformization \((\Omega_2, G_2, P_2:\Omega_1\to S)\) if there is a holomorphic map \(Q: \Omega_1\to \Omega_2\) such that \(P_1=P_2\circ Q\). The main theorem of this paper states that the lowest uniformizations of a pure compact Klein surface are exactly the extended Schottky uniformizations. The corresponding result for closed Riemann surfaces is proved by \textit{B. Maskit} [Ann. Math. (2) 81, 341--355 (1965; Zbl 0151.33003)] and for pure closed Klein surfaces by the author and \textit{B. Maskit} [Contemp. Math. 397, 145--152 (2006; Zbl 1103.30025)]. Thus in this paper the author ``completes the story'' of determining the lowest uniformizations of compact Klein surfaces.