On a theorem of Scott and Swarup

Abstract

Let 1 → H → G → Z → 1 be an exact sequence of hyperbolic groups induced by an automorphism Φ of the free group H. Let H₁(⊂ H) be a finitely generated distorted subgroup of G. Then there exist N > 0 and a free factor K of H such that the conjugacy class of K is preserved by Φ^N and H₁ contains a finite index subgroup of a conjugate of K. This is an analog of a Theorem of Scott and Swarup for surfaces in hyperbolic 3-manifolds.