A Non-quasiconvex Subgroup of a Hyperbolic Group with an Exotic Limit Set

We construct an example of a torsion free freely indecomposable finitely presented nonquasiconvex subgroup H of a word hyperbolic group G such that the limit set of G is not the limit set of a quasiconvex subgroup of G. In particular, this gives a counterexample to the conjecture of G.Swarup that a finitely presented one-ended subgroup of a word hyperbolic group is quasiconvex if and only if it has finite index in its virtual normalizer.