Dynamical properties of convex cocompact actions in projective space

Abstract

We give a dynamical characterization of convex cocompact group actions on properly convex domains in projective space in the sense of Danciger–Guéritaud–Kassel: we show that convex cocompactness in is equivalent to an expansion property of the group about its limit set, occurring in different Grassmannians. As an application, we give a sufficient and necessary condition for convex cocompactness for groups that are hyperbolic relative to a collection of convex cocompact subgroups. We show that convex cocompactness in this situation is equivalent to the existence of an equivariant homeomorphism from the Bowditch boundary to the quotient of the limit set of the group by the limit sets of its peripheral subgroups.

Convex co‐compact representations of 3‐manifold groups

“…(1) Boundary maps: A version of Theorem 1.9 is now known to be true for every relatively hyperbolic (naive) convex co-compact group. In particular, very shortly after this paper appeared on the arXiv, Weisman [51] posted a preprint to the arXiv showing that if Γ ⩽ Aut(Ω) is a convex co-compact subgroup which is relatively hyperbolic and whose peripheral subgroups were also convex co-compact, then there is a natural equivariant homeomorphism between the Bowditch boundary and a quotient of 𝜕 i  Ω (Γ). In recent work [30], we prove that in the above setting, the peripheral subgroups are always convex co-compact.…”