Related papers: Anosov representations and proper actions
We define the notion of affine Anosov representations of word hyperbolic groups into the affine group $\mathsf{SO}^0(n+1,n)\ltimes\mathbb{R}^{2n+1}$. We then show that a representation $\rho$ of a word hyperbolic group is affine Anosov if…
We provide characterizations of Anosov representations of word hyperbolic groups into real semisimple Lie groups in terms of the existence of equivariant limit maps on the Gromov boundary, the Cartan property and the uniform gap summation…
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 representations into rank-one Lie groups.…
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 group is not commensurable to a non-trivial…
We study the limit set of discrete subgroups arising from Anosov representations. Specially we study the limit set of discrete groups arising from strictly convex real projective structures and Anosov representations from a finitely…
We prove that any hyperbolic group acting properly discontinuously and cocompactly on a $\mathrm{CAT}(0)$ cube complex admits a projective Anosov representation into $\mathrm{SL}(d, \mathbb{R})$ for some $d$. More specifically, we show that…
Anosov representations give a higher-rank analogue of convex cocompactness in a rank-one Lie group which shares many of its good geometric and dynamical properties; geometric finiteness in rank one may be seen as a controlled weakening of…
We identify all Anosov representations of compact hyperbolic triangle reflection groups into the higher rank Lie group $\mathrm{SL}(3,\mathbb R)$. Specifically, we prove that such a representation is Anosov if and only if either it lies in…
We define a new family of discrete representations of relatively hyperbolic groups which unifies many existing definitions and examples of geometrically finite behavior in higher rank. The definition includes the relative Anosov…
We propose several common extensions of the classes of Anosov subgroups and geometrically finite Kleinian groups among discrete subgroups of semisimple Lie groups. We relativize various dynamical and coarse geometric characterizations of…
We study a notion of convex cocompactness for discrete subgroups of the projective general linear group acting (not necessarily irreducibly) on real projective space, and give various characterizations. A convex cocompact group in this…
The notion of Anosov representations has been introduced by Labourie in his study of the Hitchin component for SL(n,R). Subsequently, Anosov representations have been studied mainly for surface groups, in particular in the context of higher…
In this article, we provide a necessary and sufficient criterion for proper actions on $\mathbb{H}^{n,n-1}$ in terms of certain special Anosov representations in $\mathsf{SO}(n,n)$. Moreover, we show that affine Anosov representations of…
We study through the lens of Anosov representations the dynamical properties of reducible suspensions of linear representations of non-elementary hyperbolic groups, which are linear representations preserving and acting weakly unipotently…
This is the first in a series of two papers that develops a theory of relatively Anosov representations using the original "contracting flow on a bundle" definition of Anosov representations introduced by Labourie and Guichard-Wienhard. In…
Proper group actions are ubiquitous in mathematics and have many of the attractive features of actions of compact groups. In this survey, we discuss proper actions of Lie groups on smooth manifolds. If the group dimension is sufficiently…
We characterize groups admitting Anosov representations into $\mathsf{SL}(3,\mathbb R)$, projective Anosov representations into $\mathsf{SL}(4,\mathbb R)$, and Borel Anosov representations into $\mathsf{SL}(4,\mathbb R)$. More generally, we…
Anosov representations were introduced by F. Labourie [18] for fundamental groups of closed negatively curved surfaces, and generalized by O. Guichard and A. Wienhard [19] to representations of arbitrary Gromov hyperbolic groups into real…
We investigate representations of Coxeter groups into $\mathrm{GL}(n,\mathbb{R})$ as geometric reflection groups which are convex cocompact in the projective space $\mathbb{P}(\mathbb{R}^n)$. We characterize which Coxeter groups admit such…
In 1990, Hitchin's proved a component of the space of representations of a surface group in SL(n,R) is homeomorphic to a ball. For n=2,3 this component has been identified with the holonomies of geometric structures (hyperbolic for n=2, or…