2 \pi-grafting and complex projective structures, I

TL;DR

This paper investigates the Grafting Conjecture for complex projective structures on surfaces, proving it holds locally in the space of geodesic laminations and setting the stage for a proof in the generic case.

Contribution

It demonstrates that 2π-graftings can produce all projective structures with fixed holonomy locally in Thurston coordinates, advancing understanding of the Grafting Conjecture.

Findings

Gallo, Kapovich, and Marden's Grafting Conjecture holds locally in the space of geodesic laminations.

A natural projection of projective structures into the space of geodesic laminations is used.

The results pave the way to prove the conjecture for generic holonomy in future work.

Abstract

Let $S$ be a closed oriented surface of genus at least two. Gallo, Kapovich, and Marden asked if 2\pi-graftings produce all projective structures on $S$ with arbitrarily fixed holonomy (Grafting Conjecture). In this paper, we show that the conjecture holds true "locally" in the space $GL$ of geodesic laminations on $S$ via a natural projection of projective structures on $S$ into $GL$ in the Thurston coordinates. In the sequel paper, using this local solution, we prove the conjecture for generic holonomy.