Some Graftings of Complex Projective Structures with Schottky Holonomy

TL;DR

This paper explores the structure and connectivity of graphs formed by grafting complex projective structures with Schottky holonomy, providing formulas for grafting compositions and demonstrating the construction of infinitely many related structures.

Contribution

It introduces explicit formulas for grafting compositions in the Schottky case and shows how to construct numerous projective structures connected by grafting.

Findings

The graph of unmarked structures with Schottky holonomy is connected.

Explicit formulas for grafting compositions are derived.

Infinite ways to connect pairs of structures via grafting are demonstrated.

Abstract

Let $\mathcal{G}^*(S,\rho)$ be the graph whose vertices are marked complex projective structures with holonomy $\rho$ and whose edges are graftings from one vertex to another. If $\rho$ is quasi-Fuchsian, a theorem of Goldman implies that $\mathcal{G}^*(S,\rho)$ is connected. If $\rho(\pi_1(S))$ is a Schottky group Baba has shown that $\mathcal{G}(S,\rho)$ (the corresponding graph for unmarked structures) is connected. For the case that $\rho(\pi_1(S))$ is a Schottky group, this paper provides formulae for the composition of graftings in a basic setting. Using these formulae, one can construct an infinite number of (standard) projective structures which can be grafted to a common structure. Furthermore, one can construct pairs of projective structures which can be connected by grafting in an infinite number of ways.