Branched projective structures with Fuchsian holonomy

Calsamiglia, Gabriel; Deroin, Bertrand; Francaviglia, Stefano

doi:10.2140/gt.2014.18.379

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Branched projective structures with Fuchsian holonomy

Gabriel Calsamiglia, Bertrand Deroin and Stefano Francaviglia

Geometry & Topology 18 (2014) 379–446

DOI: 10.2140/gt.2014.18.379

Abstract

We prove that if $S$ is a closed compact surface of genus $g \geq 2$ , and if $ρ : π_{1} (S) \to PSL (2, ℂ)$ is a quasi-Fuchsian representation, then the space $ℳ_{k, ρ}$ of branched projective structures on $S$ with total branching order $k$ and holonomy $ρ$ is connected, for $k > 0$ . Equivalently, two branched projective structures with the same quasi-Fuchsian holonomy and the same number of branch points are related by a movement of branch points. In particular grafting annuli are obtained by moving branch points. In the appendix we give an explicit atlas for $ℳ_{k, ρ}$ for non-elementary representations $ρ$ . It is shown to be a smooth complex manifold modeled on Hurwitz spaces.

Keywords

projective structures, fuchsian holonomy, moduli spaces

Mathematical Subject Classification 2010

Primary: 30F35, 57M20

Secondary: 53A30, 14H15

References

Publication

Received: 16 April 2012

Accepted: 30 May 2013

Published: 29 January 2014

Proposed: Benson Farb

Seconded: Yasha Eliashberg, Jean-Pierre Otal

Authors

Gabriel Calsamiglia
Instituto de Matemática Universidade Federal Fluminense Rua Mário Santos Braga s/n 24020-140, Niterói Brazil

Bertrand Deroin
Département de mathématiques d’Orsay Université Paris 11 91405 Orsay Cedex France

Stefano Francaviglia
Dipartimento di Matematica Università di Bologna P.zza Porta S. Donato 5 40126 Bologna Italy

Volume 18, issue 1 (2014)