The oriented graph of multi-graftings in the Fuchsian case

TL;DR

This paper investigates the structure of the graph formed by grafting operations on complex projective structures with Fuchsian holonomy, proving its connectedness and computing its diameter.

Contribution

It establishes the connectedness and calculates the diameter of the oriented grafting graph for exotic complex projective structures with Fuchsian holonomy.

Findings

The grafting graph is connected.

The diameter of the grafting graph is explicitly calculated.

The results apply to structures with Fuchsian holonomy.

Abstract

We prove the connectedness and calculate the diameter of the oriented graph of graftings associated to exotic complex projective structures on a compact surface S with a given holonomy representation of Fuchsian type. The oriented graph of graftings is the graph whose vertices are the equivalence classes of marked CP^1-structures on S with a given fixed holonomy, and there is an oriented edge between two structures if the second is obtained from the first by grafting.