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 cannot be extended to splittings over more complicated subgroups.
Cite this article
Thomas Delzant, Panos Papasoglu, Codimension one subgroups and boundaries of hyperbolic groups. Groups Geom. Dyn. 4 (2010), no. 3, pp. 533–548DOI 10.4171/GGD/94