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 Teichmüller 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 Γ→G we explicitly construct open subsets of compact G-spaces, on which Γ acts properly discontinuously and with compact quotient.

As a consequence we show that higher Teichmüller 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.

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Correspondence to Anna Wienhard.

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A.W. was partially supported by the National Science Foundation under agreement No. DMS-1065919 and DMS-0846408. O.G. was partially supported by the Agence Nationale de la Recherche under ANR’s projects Repsurf (ANR-06-BLAN-0311) and ETTT (ANR-09-BLAN-0116-01) and by the National Science Foundation under agreement No. DMS-0635607.

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Guichard, O., Wienhard, A. Anosov representations: domains of discontinuity and applications. Invent. math. 190, 357–438 (2012). https://doi.org/10.1007/s00222-012-0382-7

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Keywords

  • Hyperbolic groups
  • Surface groups
  • Hitchin component
  • Maximal representations
  • Anosov representations
  • Higher Teichmüller spaces
  • Compact Clifford–Klein forms
  • Discrete subgroups of Lie groups
  • Convex cocompact subgroups
  • Quasi-isometric embedding