Suppose we have an arbitrary finite collection $X = (x_i)_{i = 1} ^ n$ of the boundary sphere $\partial D$ where $D$ represents the Poincare disk. Does there exist a (discrete?) subgroup $G$ of the isometry group of $D$ whose limiting set $\Lambda(G)$ is $X$? I know $G$ is not elementary as long as $n > 2$.
I am particularly interested in the case where $X$ consists of the $n$th degree roots of unity.