. 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. Contents 1. Introduction 184 2. Some Definitions and Notations 185 3. The Proofs 187 References 195 1. Introduction A subgroup H of a word hyperbolic group G is quasiconvex (or rational) in G if for any finite generating set A of G there is ffl ? 0 such that every geodesic in the Cayley graph \Gamma(G; A) of G with both endpoints in H is contained in ffl-neighborhood of H . The notion of a quasiconvex subgroup corresponds, roughly speaking, to that of geometric finiteness in the theory of classical hyperbolic groups (see [Swa], [KS], [Pi]). Quas..