Minimality of the Kleinian limit set

Every nonempty closed boundary subset invariant under a non-elementary Kleinian group contains the Kleinian limit set. Such a subset cannot be a singleton. Its hyperbolic convex hull contains an interior point, whose orbit lies in the hull; in the Klein ball model the boundary of this hull is contained in the original closed boundary subset.

The Kleinian limit set of an infinite subgroup is nonempty. If it is normal, its limit set is invariant under the ambient group by base-point independence and conjugation. Minimality of the Kleinian limit set gives one inclusion, and subgroup containment gives the other. Neither properness nor finite generation of the subgroup is needed.