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 $\mathbb{R}{\mathrm{P}}^d$ 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.

中文翻译：

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射影空间中凸协紧作用的动力学性质

我们给出了 Danciger-Guéritaud-Kassel 意义上的射影空间中适当凸域上的凸余紧群作用的动力学特征：我们证明了$\mathbb{右}{\mathrm{磷}}^d$相当于群关于其极限集的展开性质，发生在不同的格拉斯曼函数中。作为一个应用，我们给出了相对于凸余紧子群的集合是双曲线的群的凸余紧性的充分必要条件。我们证明，在这种情况下，凸协紧性等价于从鲍迪奇边界到群的极限集除以其外围子群的极限集的商的等变同胚的存在。