# Real Structures on Marked Schottky Space

## TL;DR

This paper characterizes all the real structures on the marked Schottky space, a complex manifold parametrizing quasiconformal deformations of Schottky groups, and describes their real parts.

## Contribution

It provides a complete classification of real structures on the marked Schottky space up to holomorphic automorphisms.

## Key findings

- Classification of all real structures on ${
m MS}_g$
- Description of the real parts of these structures
- Holomorphic automorphisms relating different real structures

## 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.