Let be a surface group
of higher genus. Let
be a discrete faithful representation with image contained in the natural embedding
of in
as a
group preserving a point and a disjoint projective line in the projective plane. We prove
that is
–Anosov
(following the terminology of Labourie [Invent. Math. 165 (2006) 51-114]), where
is
the frame bundle. More generally, we prove that all the deformations
studied in our paper [Geom. Topol. 5 (2001) 227-266] are
–Anosov. As a
corollary, we obtain all the main results of this paper and extend them to any small deformation
of , not
necessarily preserving a point or a projective line in the projective space: in particular, there is a
–invariant solid torus
in the flag variety.
The quotient space
is a flag manifold, naturally equipped with two
–dimensional
transversely projective foliations arising from the projections
of the flag variety on the projective plane and its dual; if
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
preserves
a point or a projective line in the projective plane. All these results hold for any
–Anosov
representation which is not quasi-Fuchsian, ie, does not preserve a strictly convex
domain in the projective plane.
