Some Graftings of Complex Projective Structures with Schottky Holonomy

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.