Volume 14, issue 1 (2010)

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A Schottky decomposition theorem for complex projective structures

Shinpei Baba

Geometry & Topology 14 (2010) 117–151
Abstract

Let $S$ be a closed orientable surface of genus at least two, and let $C$ be an arbitrary (complex) projective structure on $S$. We show that there is a decomposition of $S$ into pairs of pants and cylinders such that the restriction of $C$ to each component has an injective developing map and a discrete and faithful holonomy representation. This decomposition implies that every projective structure can be obtained by the construction of Gallo, Kapovich, and Marden. Along the way, we show that there is an admissible loop on $\left(S,C\right)$, along which a grafting can be done.

Keywords
complex projective structure, bending map, measured lamination
Mathematical Subject Classification 2000
Primary: 57M50
Secondary: 30F40, 53A30
Publication
Accepted: 7 September 2009
Preview posted: 10 October 2009
Published: 2 January 2010
Proposed: David Gabai
Seconded: Walter Neumann, Benson Farb
Authors
 Shinpei Baba Mathematisches Institut Universität Bonn 53115 Bonn Germany