## Geometry & Topology

### Branched projective structures with Fuchsian holonomy

#### Abstract

We prove that if $S$ is a closed compact surface of genus $g≥2$, and if $ρ:π1(S)→ PSL(2,ℂ)$ is a quasi-Fuchsian representation, then the space $ℳk,ρ$ 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 $ℳk,ρ$ for non-elementary representations $ρ$. It is shown to be a smooth complex manifold modeled on Hurwitz spaces.

#### Article information

Source
Geom. Topol., Volume 18, Number 1 (2014), 379-446.

Dates
Accepted: 30 May 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732726

Digital Object Identifier
doi:10.2140/gt.2014.18.379

Mathematical Reviews number (MathSciNet)
MR3159165

Zentralblatt MATH identifier
1286.30031

#### Citation

Calsamiglia, Gabriel; Deroin, Bertrand; Francaviglia, Stefano. Branched projective structures with Fuchsian holonomy. Geom. Topol. 18 (2014), no. 1, 379--446. doi:10.2140/gt.2014.18.379. https://projecteuclid.org/euclid.gt/1513732726

#### References

• E Arbarello, M Cornalba, P A Griffiths, Geometry of algebraic curves, Vol. II, Grundl. Math. Wissen. 268, Springer, Heidelberg (2011)
• S Baba, $2\pi$–graftings and complex projective structures, I
• L Bers, Spaces of Riemann surfaces as bounded domains, Bull. Amer. Math. Soc. 66 (1960) 98–103
• S Choi, H Lee, Geometric structures on manifolds and holonomy-invariant metrics, Forum Math. 9 (1997) 247–256
• G Faltings, Real projective structures on Riemann surfaces, Compositio Math. 48 (1983) 223–269
• D Gallo, M Kapovich, A Marden, The monodromy groups of Schwarzian equations on closed Riemann surfaces, Ann. of Math. 151 (2000) 625–704
• É Ghys, Déformations des structures complexes sur les espaces homogènes de ${\rm SL}(2,\mathbf C)$, J. Reine Angew. Math. 468 (1995) 113–138
• W Goldman, Projective structures with Fuchsian holonomy, J. Differential Geom. 25 (1987) 297–326
• A Grothendieck, Techniques de construction en géométrie analytique, X: Construction de l'espace de Teichmüller, from: “Familles d'espaces complexes et fondements de la géométrie analytique”, Sem. H Cartan 13, ENS, Paris (1962)
• R C Gunning, Special coordinate coverings of Riemann surfaces, Math. Ann. 170 (1967) 67–86
• J Harris, I Morrison, Moduli of curves, Graduate Texts in Mathematics 187, Springer, New York (1998)
• D A Hejhal, Monodromy groups and linearly polymorphic functions, Acta Math. 135 (1975) 1–55
• P Hubert, H Masur, T Schmidt, A Zorich, Problems on billiards, flat surfaces and translation surfaces, from: “Problems on mapping class groups and related topics”, (B Farb, editor), Proc. Sympos. Pure Math. 74, Amer. Math. Soc. (2006) 233–243
• A T Huckleberry, G A Margulis, Invariant analytic hypersurfaces, Invent. Math. 71 (1983) 235–240
• A Hurwitz, Ueber Riemann'sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891) 1–60
• K Kodaira, Complex manifolds and deformation of complex structures, Grundl. Math. Wissen. 283, Springer, New York (1986)
• M Kontsevich, A Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003) 631–678
• F Liu, B Osserman, The irreducibility of certain pure-cycle Hurwitz spaces, Amer. J. Math. 130 (2008) 1687–1708
• F Loray, J V Pereira, Transversely projective foliations on surfaces: Existence of minimal form and prescription of monodromy, Internat. J. Math. 18 (2007) 723–747
• R Mandelbaum, Branched structures on Riemann surfaces, Trans. Amer. Math. Soc. 163 (1972) 261–275
• R Mandelbaum, Branched structures and affine and projective bundles on Riemann surfaces, Trans. Amer. Math. Soc. 183 (1973) 37–58
• R Mandelbaum, Unstable bundles and branched structures on Riemann surfaces, Math. Ann. 214 (1975) 49–59
• B Maskit, On a class of Kleinian groups, Ann. Acad. Sci. Fenn. Ser. A I 442 (1969)
• D V Mathews, Hyperbolic cone-manifold structures with prescribed holonomy, I: Punctured tori, Geom. Dedicata 152 (2011) 85–128
• D V Mathews, Hyperbolic cone-manifold structures with prescribed holonomy, II: Higher genus, Geom. Dedicata 160 (2012) 15–45
• B A Scárdua, Transversely affine and transversely projective holomorphic foliations, Ann. Sci. École Norm. Sup. 30 (1997) 169–204
• D Sullivan, W Thurston, Manifolds with canonical coordinate charts: Some examples, Enseign. Math. 29 (1983) 15–25
• S P Tan, Branched $\mathbf C{\rm P}\sp 1$–structures on surfaces with prescribed real holonomy, Math. Ann. 300 (1994) 649–667
• F Touzet, Sur les feuilletages holomorphes transversalement projectifs, Ann. Inst. Fourier $($Grenoble$)$ 53 (2003) 815–846