We construct an example of a torsion free freely indecomposable finitely presented non-quasiconvex subgroup H of a word hyperbolic group G such that the limit set of H 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.

This research is supported by an Alfred P. Sloan Doctoral Dissertation Fellowship