# Convex co-compact actions of relatively hyperbolic groups.

@article{Islam2019ConvexCA, title={Convex co-compact actions of relatively hyperbolic groups.}, author={Mitul Islam and Andrew M. Zimmer}, journal={arXiv: Geometric Topology}, year={2019} }

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 sufficient conditions for the group to be relatively hyperbolic in terms of the geometry of the convex domain. This answers a question of Danciger-Gueritaud-Kassel and is analogous to a result of Hruska-Kleiner for ${\rm CAT}(0)$ spaces.

## 7 Citations

### The structure of relatively hyperbolic groups in convex real projective geometry

- Mathematics
- 2022

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 groups with one dimensional boundary faces

- Mathematics
- 2021

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…

### Convex co-compact representations of 3-manifold groups.

- Mathematics
- 2020

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…

### Dynamical properties of convex cocompact actions in projective space.

- Mathematics
- 2020

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

- Mathematics
- 2019

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…

### Patterson--Sullivan densities in convex projective geometry

- Mathematics
- 2021

For any rank-one convex projective manifold with a compact convex core, we prove that there exists a unique probability measure of maximal entropy on the set of unit tangent vectors whose geodesic is…

### The boundary of rank-one divisible convex sets

- Mathematics
- 2021

We prove that for any non-symmetric irreducible divisible convex set, the proximal limit set is the full projective boundary.

## References

SHOWING 1-10 OF 39 REFERENCES

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

- MathematicsJournal of the London Mathematical Society
- 2020

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…

### Convex cocompact actions in real projective geometry

- Mathematics
- 2017

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…

### Convex cocompactness in pseudo-Riemannian hyperbolic spaces

- Mathematics
- 2017

Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact…

### Convex projective generalized dehn filling

- MathematicsAnnales scientifiques de l'École normale supérieure
- 2020

For $d=4, 5, 6$, we exhibit the first examples of complete finite volume hyperbolic $d$-manifolds $M$ with cusps such that infinitely many $d$-orbifolds $M_{m}$ obtained from $M$ by generalized Dehn…

### Convex projective structures on nonhyperbolic three-manifolds

- Mathematics
- 2018

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…

### Convexes divisibles IV : Structure du bord en dimension 3

- Mathematics
- 2006

AbstractDivisible convex sets IV: Boundary structure in dimension 3
Let Ω be an indecomposable properly convex open subset of the real projective 3-space which is divisible i.e. for which there…

### Convex projective structures on Gromov-Thurston manifolds

- Mathematics
- 2006

We consider Gromov–Thurston examples of negatively curved n-manifolds which do not admit metrics of constant sectional curvature. We show that for each n ≥ 4 some of the Gromov–Thurston manifolds…