Anosov representations: Domains of discontinuity and applications

Abstract

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 Teichmueller spaces, and for lattices in SO(1,n). In this article we extend the notion of Anosov representations to representations of arbitrary word hyperbolic groups and start the systematic study of their geometric properties. In particular, given an Anosov representation of $\Gamma$ into G we explicitly construct open subsets of compact G-spaces, on which $\Gamma$ acts properly discontinuously and with compact quotient. As a consequence we show that higher Teichmueller spaces parametrize locally homogeneous geometric structures on compact manifolds. We also obtain applications regarding (non-standard) compact Clifford-Klein forms and compactifications of locally symmetric spaces of infinite volume.

63 pages, accepted for publication in Inventiones Mathematicae

Cited by in corpus (39)

• Anosov representations and proper actions
• Dynamics on flag manifolds: domains of proper discontinuity and cocompactness
• Hyperconvex representations and exponential growth
• The geometry of maximal representations of surface groups into SO(2,n)
• Parametrizing Hitchin components
• Anosov groups: local mixing, counting, and equidistribution
• Topological restrictions on Anosov representations
• The geometry of maximal components of the PSp(4,R) character variety
• Local mixing of one-parameter diagonal flows on Anosov homogeneous spaces
• Finsler bordifications of symmetric and certain locally symmetric spaces
• Degeneration of Hitchin representations along internal sequences
• Patterson-Sullivan theory for Anosov subgroups
• SO(p,q)-Higgs bundles and higher Teichmüller components
• A combination theorem for Anosov subgroups
• Collar lemma for Hitchin representations
• Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations
• Affine Anosov representations and proper actions
• Amalgam Anosov representations
• Positivity and representations of surface groups
• Schottky groups and maximal representations
• Margulis Multiverse: Infinitesimal Rigidity, Pressure Form and Convexity
• On Borel Anosov subgroups of $\mathrm{SL}(d,\mathbb{R})$
• Morse actions of discrete groups on symmetric spaces: Local-to-global principle
• Goldman Algebra, Opers and the Swapping Algebra
• Rigidity of convex divisible domains in flag manifolds
• Cubulated hyperbolic groups admit Anosov representations
• From curves to currents
• The Pressure Metric on the Margulis Multiverse
• Length functions of Hitchin representations
• Temperedness of locally symmetric spaces: The product case
• Absence of principal eigenvalues for higher rank locally symmetric spaces
• Anosov representations acting on homogeneous spaces: domains of discontinuity
• Maximal and Borel Anosov representations in $Sp(4,\mathbb{R})$
• On the limit set of a spherical CR uniformization
• Nearly geodesic immersions and domains of discontinuity
• Cataclysms for Anosov representations
• On Character Variety of Anosov Representations
• Discrete subgroups with finite Bowen-Margulis-Sullivan measure in higher rank
• Geometric structures for maximal representations and pencils