Three-dimensional Anosov flag manifolds

Abstract

Let be a surface group of higher genus. Let 0 W ! 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 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 W ! 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 0 , 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 ./n 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 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 .G; Y /-Anosov representation which is not quasi-Fuchsian, ie, does not preserve a strictly convex domain in the projective plane. 57M50

Topological invariants of Anosov representations

“…Then small deformations of ρ are (PGL(3, R), H)-Anosov, where H is the subgroup of diagonal matrices. These representations were studied in [5].…”

Domains of discontinuity for surface groups

“…For n = 3 this coincides with the domain of discontinuity defined in [1]. The construction of Section 4, applied to V = End(Λ k R n ), gives rise to a domain of discontinuity in F (R n ) which is the complement of t∈∂π 1 (Σ) L ξ k (t),ξ n−k (t) , where a flag (F 1 , .…”

Geometric Structures and Representations of Discrete Groups

“…(3) (Barbot [5] for d = 3) Let d ≥ 2. Any Fuchsian representation Γ → SL(2, R), composed with the standard embedding SL(2, R) → SL(d, R) (given by the direct sum of the standard action on R 2 and the trivial action on R d−2 ), defines a P 1 -Anosov representation Γ → G = PSL(d, R).…”