Geometry & Topology

Three-dimensional Anosov flag manifolds

Thierry Barbot

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

flag manifold Anosov representation


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

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