A flat torus theorem for convex co‐compact actions of projective linear groups

@article{Islam2020AFT,
  title={A flat torus theorem for convex co‐compact actions of projective linear groups},
  author={Mitul Islam and Andrew M. Zimmer},
  journal={Journal of the London Mathematical Society},
  year={2020},
  volume={103}
}
In this paper, we consider discrete groups in PGLd(R) acting convex co‐compactly on a properly convex domain in real projective space. For such groups, we establish an analogue of the well‐known flat torus theorem for CAT(0) spaces. 
9 Citations
Convex co-compact actions of relatively hyperbolic groups.
In this paper we consider discrete groups in ${\rm PGL}_d(\mathbb{R})$ acting convex co-compactly on a properly convex domain in real projective space. For such groups, we establish necessary and
The structure of relatively hyperbolic groups in convex real projective geometry
In this paper we prove a general structure theorem for relatively hyperbolic groups (with arbitrary peripheral subgroups) acting naive convex co-compactly on properly convex domains in real
Convex co-compact representations of 3-manifold groups.
A representation of a finitely generated group into the projective general linear group is called convex co-compact if it has finite kernel and its image acts convex co-compactly on a properly convex
A higher rank rigidity theorem for convex real projective manifolds
For convex real projective manifolds we prove an analogue of the higher rank rigidity theorem of Ballmann and Burns-Spatzier.
Codimension-1 simplices in divisible convex domains
Properly embedded simplices in a convex divisible domain $\Omega \subset \mathbb{R} \textrm{P}^d$ behave somewhat like flats in Riemannian manifolds, so we call them flats. We show that the set of
Entropy rigidity for cusped Hitchin representations
We establish an entropy rigidity theorem for Hitchin representations of all geometrically finite Fuchsian groups which generalizes a theorem of Potrie and Sambarino for Hitchin representations of
Convex co-compact groups with one dimensional boundary faces
In this paper we consider convex co-compact subgroups of the projective linear group. We prove that such a group is relatively hyperbolic with respect to a collection of virtually Abelian subgroups
Dynamical properties of convex cocompact actions in projective space.
We give a dynamical characterization of convex cocompact group actions on properly convex domains in projective space in the sense of Danciger-Gueritaud-Kassel: we show that convex cocompactness in
Rank-One Hilbert Geometries
We introduce and study the notion of rank-one Hilbert geometries and open properly convex domains in $\mathbb{P}(\mathbb{R}^{d+1})$. This is in the spirit of rank-one non-positively curved Riemannian

References

SHOWING 1-10 OF 30 REFERENCES
Convex cocompact actions in real projective geometry
We study a notion of convex cocompactness for (not necessarily irreducible) discrete subgroups of the projective general linear group acting on real projective space, and give various
Rigidity of invariant convex sets in symmetric spaces
The main result implies that a proper convex subset of an irreducible higher rank symmetric space cannot have Zariski dense stabilizer.
Groupes Convexes Cocompacts En Rang Supérieur
For higher rank semisimple Lie groups, we give an obstruction to the generalization of the notion of a discrete convex cocompact subgroup
A survey on divisible convex sets
We report without proof recent advances on the study of open properly convex subsets Ω of the real projective space which are divisible i.e. for which there exists a discrete group Γ of projective
Entropies of compact strictly convex projective manifolds
Let M be a compact manifold of dimension n with a strictly convex projective structure. We consider the geodesic flow of the Hilbert metric on it, which is known to be Anosov. We prove that its
On Convex Projective Manifolds and Cusps
Convex projective structures on nonhyperbolic three-manifolds
Y. Benoist proved that if a closed three-manifold M admits an indecomposable convex real projective structure, then M is topologically the union along tori and Klein bottles of finitely many
Around groups in Hilbert Geometry
This is survey about action of group on Hilbert geometry. It will be a chapter of the "Handbook of Hilbert geometry" edited by G. Besson, M. Troyanov and A. Papadopoulos.
Projective Anosov representations, convex cocompact actions, and rigidity
In this paper we show that many projective Anosov representations act convex cocompactly on some properly convex domain in real projective space. In particular, if a non-elementary word hyperbolic
On the Hilbert Geometry of Convex Polytopes
We survey the Hilbert geometry of convex polytopes. In particular we present two important characterisations of these geometries, the first one in terms of the volume growth of their metric balls,
...
1
2
3
...