We define the notion of affine Anosov representations of word hyperbolic groups into the affine groupWe then show that a representation ρ of a word hyperbolic group is affine Anosov if and only if its linear part Lρ is Anosov in SO 0 (n + 1, n) with respect to the stabilizer of a maximal isotropic plane and ρ(Γ) acts properly on R 2n+1 .
We study actions of discrete subgroups Γ of semi-simple Lie groups G on associated oriented flag manifolds. These are quotients G/P , where the subgroup P lies between a parabolic subgroup and its identity component. For Anosov subgroups Γ ⊂ G, we identify domains in oriented flag manifolds by removing a set obtained from the limit set of Γ, and give a combinatorial description of proper discontinuity and cocompactness of these domains. This generalizes analogous results of Kapovich-Leeb-Porti to the oriented setting. We give first examples of cocompact domains of discontinuity which are not lifts of domains in unoriented flag manifolds. These include in particular domains in oriented Grassmannians for Hitchin representations, which we also show to be nonempty. As a further application of the oriented setup, we give a new lower bound on the number of connected components of B-Anosov representations of a closed surface group into SL(n, R).
We describe a construction of Schottky type subgroups of automorphism groups of partially cyclically ordered sets. We apply this construction to the Shilov boundary of a Hermitian symmetric space and show that in this setting Schottky subgroups correspond to maximal representations of fundamental groups of surfaces with boundary. As an application, we construct explicit fundamental domains for the action of maximal representations into Sp(2n, R) on RP 2n−1 . arXiv:1609.04560v1 [math.DG]
We define the notion of affine Anosov representations of word hyperbolic groups into the affine group $\textsf {SO}^0(n+1,n)\ltimes {\mathbb {R}}^{2n+1}$. We then show that a representation $\rho $ of a word hyperbolic group is affine Anosov if and only if its linear part $\mathtt {L}_\rho $ is Anosov in $\textsf {SO}^0(n+1,n)$ with respect to the stabilizer of a maximal isotropic plane and $\rho (\Gamma )$ acts properly on $\mathbb {R}^{2n+1}$.
We define for every positive Anosov representation of a nonabelian free group into SO(2n, 2n − 1) a family of R 4n−1 -valued cocycles which induce proper affine actions on R 4n−1 . We construct fundamental domains in R 4n−1 bounded by generalized crooked planes for these affine actions, and deduce that the quotient manifolds are homeomorphic to handlebodies.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.