Abstract. In this paper we develop some of the foundations of the theory of relatively hyperbolic groups as originally formulated by Gromov. We prove the equivalence of two definitions of this notion. One is essentially that of a group admitting a properly discontinuous geometrically finite action on a proper hyperbolic space, that is, such that every limit point is either a conical limit point or a bounded parabolic point. The other is that of a group which admits a cofinite action on a connected fine hyperbolic graph. We define a graph to be "fine" if there are only finitely many circuits a given length containing any given edge, and we develop some of the properties of this notion. We show how a relatively hyperbolic group can be assumed to act on a proper hyperbolic space of a particular geometric form. We define the boundary of a relatively hyperbolic group, and show that the limit set of any geometrically finite action of the group is equivariantly homeomorphic to this boundary. This generalises a result of Tukia for geometrically finite kleinian groups. We also describe when the boundary is connected.

In this paper, we give a construction of the JSJ splitting of a one-ended hyperbolic group (in the sense of Gromov [Gr]), using the local cut point structure of the boundary. In particular, this gives the quasiisometry invariance of the splitting, as well as the annulus theorem for hyperbolic groups. The canonical nature of the splitting is also immediate from this approach.The notion of a JSJ splitting, in this context, was introduced by Sela [Se], who constructed such splittings for all (torsion-free) hyperbolic groups. They take their name from the analogy with the characteristic submanifold construction for irreducible 3-manifolds described by Jaco and Shalen [JS] and Johannson [Jo] (developing a theory outlined earlier by Waldhausen). The JSJ splitting gives a description of the set of all possible splittings of the group over two-ended subgroups, and thus tells us about the structure of the outer automorphism group.We shall take as hypothesis here the fact that the boundary is locally connected, i.e. a "Peano continuum'!. This is now known to be the case for all one-ended hyperbolic groups, from the results of [Bol], [Bo2], [L], [Sw], [Bo5], as we shall discuss shortly. This uses the fact that local connectedness is implied by the non-existence of a global cut point [BM].A generalisation of the JSJ splitting to finitely presented groups has been given by Rips and Sela [RS]. The methods of [Se] and [RS] are founded on the theory of actions on R-trees. They consider only splittings over infinite cyclic groups. It seems that their methods run into problems if one wants to consider, for example, splittings over infinite dihedral groups (see [MNS]). A more general approach to this has recently been described by Dunwoody and Sageev [DSa] using tracks on 2-complexes. Fujiwara and Papasoglu have obtained similar 146 B.H. BOWDITCH results using actions on products of trees [FP]. These methods work in a more general context than those of this paper. (They deal with splittings of finitely presented groups over "slender" subgroups.) However, one looses some information about the splitting. For example, it is not known if the splitting is quasiisometry invariant in this generality. The annulus theorem would appear to generalise, though this does not follow immediately.A proof of the latter has recently been claimed in [DSw] for finitely generated groups.(We shall return to this point later.) We shall see that for hyperbolic groups, all these results can be unified in one approach.As we have suggested, deriving the splitting from an analysis of the boundary enables us to conclude that certain topological properties of the boundary are reflected in the structure of the group. For example, we see that the splitting is non-trivial if and only if the boundary has a local cut point (see Theorem 6.2). In fact, much information about the splitting can be read off immediately, without any knowledge of how the group acts on the boundary. In the course of the analysis, we shall also give an elementary proof that the boundary...

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