Complex projective structures with Schottky holonomy

TL;DR

This paper studies complex projective structures on surfaces with Schottky holonomy, showing they can all be obtained by grafting a basic structure associated with the Schottky group.

Contribution

It demonstrates that any projective structure with a given Schottky holonomy can be constructed via grafting from a fundamental structure.

Findings

All structures with the same holonomy are obtainable by grafting.

The basic structure is associated with the Schottky group's hyperbolic handlebody.

Grafting provides a complete classification of such projective structures.

Abstract

A Schottky group in PSL(2, C) induces an open hyperbolic handlebody and its ideal boundary is a closed orientable surface S whose genus is equal to the rank of the Schottky group. This boundary surface is equipped with a (complex) projective structure and its holonomy representation is an epimorphism from pi_1(S) to the Schottky group. We will show that an arbitrary projective structure with the same holonomy representation is obtained by (2 pi-)grafting the basic structure described above.