Three-dimensional Anosov flag manifolds

Abstract

Let $\Gamma$ be a surface group of higher genus. Let ${\rho }_0:\Gamma \to PGL\left(V\right)$ be a discrete faithful representation with image contained in the natural embedding of $SL\left(2,ℝ\right)$ in $PGL\left(3,ℝ\right)$ as a group preserving a point and a disjoint projective line in the projective plane. We prove that ${\rho }_0$ is $\left(G,Y\right)$–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 $\rho :\Gamma \to PGL\left(3,ℝ\right)$ studied in our paper [Geom. Topol. 5 (2001) 227-266] are $\left(G,Y\right)$–Anosov. As a corollary, we obtain all the main results of this paper 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 $\rho \left(\Gamma \right)$–invariant solid torus $\Omega$ in the flag variety. The quotient space $\rho \left(\Gamma \right)\setminus \Omega$ 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 $\left(G,Y\right)$–Anosov representation which is not quasi-Fuchsian, ie, does not preserve a strictly convex domain in the projective plane.