Real structures on marked Schottky space

Abstract

Schottky groups are exactly those Kleinian groups providing the regular lowest planar uniformizations of closed Riemann surfaces and also the ones providing to the interior of a handlebody of a complete hyperbolic structure with injectivity radius bounded away from zero. The space parametrizing quasiconformal deformations of Schottky groups of a fixed rank $g \geq 1$ is the marked Schottky space ${\mathcal M}{\mathcal S}_{g}$; this being a complex manifold of dimension $3(g-1)$ for $g \geq 2$ and being isomorphic to the punctured unit disc for $g=1$. In this paper we provide a complete description of the real structures of ${\mathcal M}{\mathcal S}_{g}$, up to holomorphic automorphisms, together their real part.