2\pi-grafting and complex projective structures with generic holonomy

TL;DR

This paper demonstrates that for a generic holonomy representation, all complex projective structures on a surface can be obtained through (2π-)grafting, linking geometric structures to holonomy in a comprehensive way.

Contribution

It establishes that (2π-)grafting can generate all projective structures with a given generic holonomy, providing a complete characterization.

Findings

(2π-)grafting produces all structures with generic holonomy

The result applies to surfaces of genus at least two

Holonomy representations are dense in the character variety

Abstract

Let S be an oriented closed surface of genus at least two. We show that, given a generic representation in the PSL(2,C)-character variety of S, (2\pi-)graftings produce all projective structures on S with the holonomy representation.