A pair of disjoint invariant path-connected sets on the boundary of a hyperbolic 3-manifold canonically yields a universal circle with invariant laminations.
Bowditch,A topological characterisation of hyperbolic groups, JAMS 11 (1998), no
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A pair of disjoint invariant path-connected sets on the boundary of a hyperbolic 3-manifold canonically yields a universal circle with invariant laminations.