Some Graftings of Complex Projective Structures with Schottky Holonomy

Abstract

Let ${\mathcal{G}}^{\ast }(S,\rho )$ 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 ${\mathcal{G}}^{\ast }(S,\rho )$ 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.

32 pages, 10 figures

Cited by in corpus (1)

• The oriented graph of multi-graftings in the Fuchsian case