  The intersection pattern of the translates of the limit set of a quasi-convex
subgroup of a hyperbolic group can be coded in a natural incidence graph, which
suggests connections with the splittings of the ambient group. A similar
incidence graph exists for any subgroup of a group. We show that the
disconnectedness of this graph for codimension one subgroups leads to
splittings. We also reprove some results of Peter Kropholler on splittings of
groups over malnormal subgroups and variants of them.
