Domains of discontinuity in oriented flag manifolds

Abstract

We study actions of discrete subgroups Γ $\Gamma$ of semi-simple Lie groups G $G$ on associated oriented flag manifolds. These are quotients G / P $G/P$ , where the subgroup P $P$ lies between a parabolic subgroup and its identity component. For Anosov subgroups Γ ⊂ G $\Gamma \subset G$ , we identify domains in oriented flag manifolds by removing a set obtained from the limit set of Γ $\Gamma$ , 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 $B$ -Anosov representations of a closed surface group into SL ( n , R ) $\operatorname{SL}(n,\mathbb {R})$ .