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Three-dimensional Anosov flag manifolds

Thierry Barbot

Geometry & Topology 14 (2010) 153–191

Let Γ be a surface group of higher genus. Let ρ0: Γ PGL(V ) be a discrete faithful representation with image contained in the natural embedding of SL(2, ) in PGL(3, ) as a group preserving a point and a disjoint projective line in the projective plane. We prove that ρ0 is (G,Y )–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 ρ: Γ PGL(3, ) studied in our paper [Geom. Topol. 5 (2001) 227-266] are (G,Y )–Anosov. As a corollary, we obtain all the main results of this paper and extend them to any small deformation of ρ0, not necessarily preserving a point or a projective line in the projective space: in particular, there is a ρ(Γ)–invariant solid torus Ω in the flag variety. The quotient space ρ(Γ)Ω 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 ρ 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 ρ 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.

flag manifold, Anosov representation
Mathematical Subject Classification 2000
Primary: 57M50
Received: 25 April 2007
Revised: 14 October 2008
Accepted: 17 August 2009
Preview posted: 10 October 2009
Published: 2 January 2010
Proposed: Martin Bridson
Seconded: Danny Calegari, David Gabai
Thierry Barbot
EA 2151
Université d’Avignon
33, rue Louis Pasteur
F-84 000 Avignon