Olivier Guichard

Anosov representations: domains of discontinuity and applications

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.

Anosov representations and proper actions

We establish several characterizations of Anosov representations of word hyperbolic groups into real reductive Lie groups, in terms of a Cartan projection or Lyapunov projection of the Lie group. Using a properness criterion of Benoist and Kobayashi, we derive applications to proper actions on homogeneous spaces of reductive groups.

Topological invariants of Anosov representations

We define new topological invariants for Anosov representations and study them in detail for maximal representations of the fundamental group of a closed oriented surface Σ into the symplectic group Sp (2n, R). In particular we show that the invariants distinguish connected components of the space of symplectic maximal representations other than Hitchin components. Since the invariants behave naturally with respect to the action of the mapping class group of Σ, we obtain from this the number of components of the quotient by the mapping class group action. For specific symplectic maximal representations we compute the invariants explicitly. This allows us to construct nice model representations in all connected components. The construction of model representations is of particular interest for Sp (4, R), because in this case there are −1−χ(Σ) connected components in which all representations are Zariski dense and no model representations have been known so far. Finally, we use the model representations to draw conclusions about the holonomy of symplectic maximal representations.

Compactification of certain Clifford-Klein forms of reductive homogeneous spaces

We describe smooth compactifications of certain families of reductive homogeneous spaces such as group manifolds for classical Lie groups, or pseudo-Riemannian analogues of real hyperbolic spaces and their complex and quaternionic counterparts. We deduce compactifications of Clifford-Klein forms of these homogeneous spaces, namely quotients by discrete groups Gamma acting properly discontinuously, in the case that Gamma is word hyperbolic and acts via an Anosov representation. In particular, these Clifford-Klein forms are topologically tame.