  A combinatorial condition is obtained for when immersed or embedded
incompressible surfaces in compact 3-manifolds with tori boundary components
remain incompressible after Dehn surgery. A combinatorial characterisation of
hierarchies is described. A new proof is given of the topological rigidity
theorem of Hass and Scott for 3-manifolds containing immersed incompressible
surfaces, as found in cubings of non-positive curvature.
