Geometry & Topology

Three-dimensional Anosov flag manifolds

Thierry Barbot

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Abstract

Let Γ be a surface group of higher genus. Let ρ0:Γ PGL(V) be a discrete faithful representation with image contained in the natural embedding of SL(2,) in PGL(3,) as a group preserving a point and a disjoint projective line in the projective plane. We prove that ρ0 is (G,Y)–Anosov (following the terminology of Labourie [Invent. Math. 165 (2006) 51-114]), where Y is the frame bundle. More generally, we prove that all the deformations ρ:Γ PGL(3,) studied in our paper [Geom. Topol. 5 (2001) 227-266] are (G,Y)–Anosov. As a corollary, we obtain all the main results of this paper and extend them to any small deformation of ρ0, not necessarily preserving a point or a projective line in the projective space: in particular, there is a ρ(Γ)–invariant solid torus Ω in the flag variety. The quotient space ρ(Γ)Ω is a flag manifold, naturally equipped with two 1–dimensional transversely projective foliations arising from the projections of the flag variety on the projective plane and its dual; if ρ is strongly irreducible, these foliations are not minimal. More precisely, if one of these foliations is minimal, then it is topologically conjugate to the strong stable foliation of a double covering of a geodesic flow, and ρ preserves a point or a projective line in the projective plane. All these results hold for any (G,Y)–Anosov representation which is not quasi-Fuchsian, ie, does not preserve a strictly convex domain in the projective plane.

Article information

Source
Geom. Topol., Volume 14, Number 1 (2010), 153-191.

Dates
Received: 25 April 2007
Revised: 14 October 2008
Accepted: 17 August 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732174

Digital Object Identifier
doi:10.2140/gt.2010.14.153

Mathematical Reviews number (MathSciNet)
MR2578303

Zentralblatt MATH identifier
1177.57011

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds

Keywords
flag manifold Anosov representation

Citation

Barbot, Thierry. Three-dimensional Anosov flag manifolds. Geom. Topol. 14 (2010), no. 1, 153--191. doi:10.2140/gt.2010.14.153. https://projecteuclid.org/euclid.gt/1513732174


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References

  • V Bangert, Minimal geodesics, Ergodic Theory Dynam. Systems 10 (1990) 263–286
  • T Barbot, Flag structures on Seifert manifolds, Geom. Topol. 5 (2001) 227–266
  • T Barbot, Plane affine geometry and Anosov flows, Ann. Sci. École Norm. Sup. $(4)$ 34 (2001) 871–889
  • Y Benoist, Automorphismes des cônes convexes, Invent. Math. 141 (2000) 149–193
  • R Bowen, B Marcus, Unique ergodicity for horocycle foliations, Israel J. Math. 26 (1977) 43–67
  • M Brunella, Expansive flows on Seifert manifolds and on torus bundles, Bol. Soc. Brasil. Mat. $($N.S.$)$ 24 (1993) 89–104
  • S Choi, W M Goldman, Convex real projective structures on closed surfaces are closed, Proc. Amer. Math. Soc. 118 (1993) 657–661
  • A Fathi, F Laudenbach, V Poenaru (editors), Travaux de Thurston sur les surfaces, Astérisque 66–67, Soc. Math. France, Paris (1979) Séminaire Orsay, With an English summary
  • É Ghys, Flots d'Anosov dont les feuilletages stables sont différentiables, Ann. Sci. École Norm. Sup. $(4)$ 20 (1987) 251–270
  • V L Ginzburg, B Z Gürel, A $C\sp 2$–smooth counterexample to the Hamiltonian Seifert conjecture in $\Bbb R\sp 4$, Ann. of Math. $(2)$ 158 (2003) 953–976
  • W M Goldman, Geometric structures on manifolds and varieties of representations, from: “Geometry of group representations (Boulder, CO, 1987)”, (W M Goldman, A R Magid, editors), Contemp. Math. 74, Amer. Math. Soc. (1988) 169–198
  • O Guichard, Composantes de Hitchin et représentations hyperconvexes de groupes de surface, J. Differential Geom. 80 (2008) 391–431
  • G Kuperberg, A volume-preserving counterexample to the Seifert conjecture, Comment. Math. Helv. 71 (1996) 70–97
  • K Kuperberg, A smooth counterexample to the Seifert conjecture, Ann. of Math. $(2)$ 140 (1994) 723–732
  • F Labourie, Anosov flows, surface groups and curves in projective space, Invent. Math. 165 (2006) 51–114
  • F Salein, Variétés anti-de Sitter de dimension 3 possédant un champ de Killing non trivial, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 525–530
  • R Schwartz, Pappus' theorem and the modular group, Inst. Hautes Études Sci. Publ. Math. (1993) 187–206
  • W P Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Math. Ser. 35, Princeton Univ. Press (1997) Edited by S Levy
  • F Waldhausen, On irreducible $3$–manifolds which are sufficiently large, Ann. of Math. $(2)$ 87 (1968) 56–88