Codimension one subgroups and boundaries of hyperbolic groups

We construct hyperbolic groups with the following properties: The boundary of
the group has big dimension, it is separated by a Cantor set and the group does
not split. This shows that Bowditch's theorem that characterizes splittings of
hyperbolic groups over 2-ended groups in terms of the boundary can not be
extended to splittings over more complicated subgroups.