Three-dimensional Anosov flag manifolds

Let be a surface group of higher genus. Let rho(0): Gamma -> 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 rho(0) is (G, Y)-Anosov (following the terminology of Labourie [15]), where Y is the frame bundle. More generally, we prove that all the deformations rho: Gamma -> PGL (3, R) studied in our paper [2] are (G, Y)-Anosov. As a corollary, we obtain all the main results of [2] and extend them to any small deformation of rho(0), not necessarily preserving a point or a projective line in the projective space: in particular, there is a rho(Gamma)-invariant solid torus Omega in the flag variety. The quotient space rho(Gamma)\Omega 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 rho 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 rho 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.