Benoist-Hulin groups

A Benoist-Hulin group is, by definition, a subgroup $\Gamma$ of ${\rm PSL}_2(\mathbb{C})$ such that any $\Gamma$-invariant closed set consisting of Jordan curves in the space of closed subsets of the Riemann sphere that are not singletons is composed of $K$-quasicircles for some $K \ge 1$. Y.Benoist and D.Hulin showed that the full group ${\rm PSL}_2(\mathbb{C})$ is a Benoist-Hulin group. In this paper, we develop the theory of Benoist-Hulin groups and show that both uniform lattices and parabolic subgroups are Benoist-Hulin groups.