Three-dimensional anosov flag manifolds

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, R) in PGL(3, R) 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, R) 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.