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 into the symplectic group.Comment: 66 pages, several changes, some consequences adde

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.Comment: 73 pages, 4 figures; to appear in Geometry & Topolog

Noncommutative coordinates for symplectic representations

We introduce coordinates on the spaces of framed and decorated representations of the fundamental group of a surface with boundary into the symplectic group Sp(2n,R). These coordinates provide a noncommutative generalization of the parameterizations of the spaces of representations into SL(2,R) or PSL(2,R) given by Thurston, Penner, Kashaev and Fock-Goncharov. On the space of decorated symplectic representations the coordinates give a geometric realization of the noncommutative cluster-like structures introduced by Berenstein-Retakh. The locus of positive coordinates maps to the space of framed maximal representations. We use this to determine an explicit homeomorphism between the space of framed maximal representations and a quotient by the group O(n). This allows us to describe the homotopy type and, when n=2, to give an exact description of the singularities. Along the way, we establish a complete classification of pairs of nondegenerate quadratic forms.Comment: 80 page