Higher-rank generalizations of convex cocompact and geometrically finite dynamics
We study several higher-rank generalizations of the dynamical behavior of convex cocompact groups in rank-one Lie groups, in the context of both convex projective geometry and relatively hyperbolic groups. Our results include a dynamical characterization of a notion of convex cocompact projective structure due to Danciger-Guéritaud-Kassel. This generalizes a dynamical characterization of Anosov representations of hyperbolic groups. Using topological dynamics, we also define a new notion of geometrical finiteness in higher rank which generalizes previous notions of relative Anosov representation due to Kapovich-Leeb and Zhu. We prove that these “extended geometrically finite” representations are stable under certain small relative deformations, and we provide various examples coming from the theory of convex projective structures.