## Geometry & Topology

### Three-dimensional Anosov flag manifolds

Thierry Barbot

#### Abstract

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 $SL(2,ℝ)$ in $PGL(3,ℝ)$ 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 $ρ:Γ→ PGL(3,ℝ)$ 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.

#### Article information

Source
Geom. Topol., Volume 14, Number 1 (2010), 153-191.

Dates
Revised: 14 October 2008
Accepted: 17 August 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732174

Digital Object Identifier
doi:10.2140/gt.2010.14.153

Mathematical Reviews number (MathSciNet)
MR2578303

Zentralblatt MATH identifier
1177.57011

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds

#### Citation

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

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