Geometry & Topology
- Geom. Topol.
- Volume 14, Number 1 (2010), 153-191.
Three-dimensional Anosov flag manifolds
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.
Geom. Topol., Volume 14, Number 1 (2010), 153-191.
Received: 25 April 2007
Revised: 14 October 2008
Accepted: 17 August 2009
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 57M50: Geometric structures on low-dimensional manifolds
Barbot, Thierry. Three-dimensional Anosov flag manifolds. Geom. Topol. 14 (2010), no. 1, 153--191. doi:10.2140/gt.2010.14.153. https://projecteuclid.org/euclid.gt/1513732174