LetH be a hyperbolic normal subgroup of infinite index in a hyperbolic group G. It follows from work of Rips and Sela [16] (see below), that H has to be a free product of free groups and surface groups if it is torsion-free. From [14], the quotient group Q is hyperbolic and contains a free cyclic subgroup. This gives rise to a hyperbolic automorphism [2] of H . By iterating this automorphism, and scaling the Cayley graph of H , we get a sequence of actions of H on δi-hyperbolic metric spaces, where δi → 0 as i → ∞. From this, one can extract a subsequence converging to a small isometric action on a 0-hyperbolic metric space, i.e. an R-tree. By the JSJ splitting of Rips and Sela [16], [17], the outer automorphism group of H is generated by internal automorphisms. One notes further, that a hyperbolic automorphism cannot preserve any splitting over cyclic subgroups and that the limiting action is in fact free. Hence, by a theorem of Rips [16], H has to be a free product of free groups and surface groups if it is torsion-free. Thus the collection of normal subgroups possible is limited. However, the class of groups G can still be fairly large. Examples can be found in [3], [5] and [13]. For the purposes of this paper we choose a finite generating set of G that contains a finite generating set of H . Let ΓG and ΓH be the Cayley graphs of G, H with respect to these generating sets. There is a continuous proper embedding i of ΓH into ΓG. Every hyperbolic group admits a compactification of its Cayley graph by adjoining the Gromov boundary consisting of