Geometry & Topology

Espace des modules marqués des surfaces projectives convexes de volume fini

Ludovic Marquis

Full-text: Open access

Abstract

Cet article est la suite de l’article [arXiv :0902.3143] dans lequel l’auteur caractérisait le fait d’être de volume fini pour une surface projective convexe. On montre ici que l’espace des modules βf(Σg,p) des structures projectives convexes de volume fini sur la surface βf(Σg,p) de genre g à p pointes est homéomorphe à 16g16+6p.

Enfin, on montre que βf(Σg,p) s’identifie à une composante connexe de l’espace des représentations du groupe fondamental de Σg,p dans SL3() qui conservent les paraboliques à conjugaison près.

This article follows the article [arXiv :0902.3143] in which the author characterizes the fact of being of finite volume for a convex projective surface. We show here that the moduli space βf(Σg,p) of convex projective structures on the surface Σg,p of genus g with p punctures is homeomorphic to 16g16+6p.

Finally, we show that βf(Σg,p) can be identified with a connected component of the space of representations of the fundamental group of Σg,p in SL3() which keep the parabolics modulo conjugation.

Article information

Source
Geom. Topol., Volume 14, Number 4 (2010), 2103-2149.

Dates
Received: 30 October 2009
Revised: 27 August 2010
Accepted: 1 August 2010
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883515

Digital Object Identifier
doi:10.2140/gt.2010.14.2103

Mathematical Reviews number (MathSciNet)
MR2740643

Zentralblatt MATH identifier
1225.32022

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds 51M10: Hyperbolic and elliptic geometries (general) and generalizations 51A05: General theory and projective geometries
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]

Keywords
convex projective geometry surface moduli space

Citation

Marquis, Ludovic. Espace des modules marqués des surfaces projectives convexes de volume fini. Geom. Topol. 14 (2010), no. 4, 2103--2149. doi:10.2140/gt.2010.14.2103. https://projecteuclid.org/euclid.gt/1513883515


Export citation

References

  • Y Benoist, Sous-groupes discrets des groupes de lie, Lecture, European Summer School in Group Theory Luminy (1997)
  • Y Benoist, Automorphismes des cônes convexes, Invent. Math. 141 (2000) 149–193
  • Y Benoist, Convexes divisibles. III, Ann. Sci. École Norm. Sup. $(4)$ 38 (2005) 793–832
  • Y Benoist, Convexes divisibles. IV. Structure du bord en dimension 3, Invent. Math. 164 (2006) 249–278
  • Y Benoist, Convexes hyperboliques et quasiisométries, Geom. Dedicata 122 (2006) 109–134
  • D Burago, Y Burago, S Ivanov, A course in metric geometry, Graduate Studies in Math. 33, Amer. Math. Soc. (2001)
  • S Choi, W M Goldman, Convex real projective structures on closed surfaces are closed, Proc. Amer. Math. Soc. 118 (1993) 657–661
  • B Colbois, C Vernicos, P Verovic, L'aire des triangles idéaux en géométrie de Hilbert, Enseign. Math. $(2)$ 50 (2004) 203–237
  • B Colbois, C Vernicos, P Verovic, Area of ideal triangles and Gromov hyperbolicity in Hilbert geometry, Illinois J. Math. 52 (2008) 319–343
  • V V Fock, A B Goncharov, Moduli spaces of convex projective structures on surfaces, Adv. Math. 208 (2007) 249–273
  • W M Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984) 200–225
  • W M Goldman, Convex real projective structures on compact surfaces, J. Differential Geom. 31 (1990) 791–845
  • W M Goldman, J J Millson, Local rigidity of discrete groups acting on complex hyperbolic space, Invent. Math. 88 (1987) 495–520
  • J-L Koszul, Déformations de connexions localement plates, Ann. Inst. Fourier $($Grenoble$)$ 18 (1968) 103–114
  • F Labourie, Flat projective structures on surfaces and cubic holomorphic differentials, Pure Appl. Math. Q. 3 (2007) 1057–1099
  • J C Loftin, Affine spheres and convex $\mathbb{RP}\sp n$–manifolds, Amer. J. Math. 123 (2001) 255–274
  • L Marquis, Surface projective convexe de volume fini A paraître dans Ann. Inst. Fourier $($Grenoble$)$