Three-dimensional Anosov flag manifolds

Let $\Gamma$ be a surface group of higher genus. Let $\rho\_0: \Gamma \to {PGL}(V)$ be a discrete faithful representation with image contained in the natural embedding of ${SL}(2, {\mathbb R})$ in ${PGL}(3, {\mathbb R})$ as a group preserving a point and a disjoint projective line in the projective plane. We prove that such a representation is $(G,Y)$-Anosov (following the terminology of \cite{labourieanosov}), where $Y$ is the frame bundle. More generally, we prove that all the deformations $\rho: \Gamma \to {PGL}(3, {\mathbb R})$ studied in \cite{barflag} are $(G,Y)$-Anosov. As a corollary, we obtain all the main results of \cite{barflag}, 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 $\rho(\Gamma)\backslash\Omega$ 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, i.e., does not preserve a strictly convex domain in the projective plane.