Some graftings of complex projective structures with Schottky holonomy
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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.
Keywords
Grafting Complex projective structures Schottky groupsMathematics Subject Classification
51M10Preview
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