Grafting Real Complex Projective Structures with Fuchsian Holonomy

Details Grafting Real Complex Projective Structures with Fuchsian Holonomy Grafting Real Complex Projective Structures with Fuchsian Holonomy

2010-12-10

arxiv.org/abs/1012.2194v1

Mathematics

Geometric Topology

31 pages, 6 figures

Scientific paper

Let $\mathcal{G}(S,\rho)$ be a graph whose vertices are 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 contractible. If $\rho$ is a quasi-Fuchsian Schottky group Baba has shown that $\mathcal{G}(S,\rho)$ is connected. We show that if $\rho$ is a quasi-Fuchsian Schottky group $\pi_1(\mathcal{G}(S,\rho))$ is not finitely generated and there are an infinte number of (standard) projective structures which can be grafted to a common structure.

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