• Corpus ID: 204800801

# 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}
}
• Published 20 October 2019
• Mathematics
• arXiv: Geometric Topology
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

## Figures from this paper

### 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.

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

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

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

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

• Mathematics
Journal 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

• Mathematics
Annales 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

### Hadamard spaces with isolated flats

• Mathematics
• 2004
We explore the geometry of nonpositively curved spaces with isolated flats, and its consequences for groups that act properly discontinuously, cocompactly, and isometrically on such spaces. We prove

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

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

### 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