On the Cut Point

We sketch a proof of the fact that the Gromov boundary of a hyperbolic group does not have a global cut point if it is connected. This implies, by a theorem of Bestvina and Mess, that the boundary is locally connected if it is connected. The object of this note is to sketch a proof of the following theorem: Theorem 1. If ? is a one ended hyperbolic group, then the Gromov boundary @? of ? does not have a global cut point. It seems that several people independently thought of using treelike structures in this context, among them Bill Grosso in an unpublished manuscript 8]. The most signiicant advances in this direction were carried out by Brian Bowditch in a brilliant series of papers ((4]-7]). We draw heavily from his work. He proved Theorem 1 in the case when ? is one ended and does not split over a two ended group 6], and in the case when ? is strongly accessible and one ended 5]. Our strategy is to relativize some of the arguments of 4], 5], use Levitt's construction 9] to obtain an R-tree on which a subgroup of ? acts isometrically, and then use a relative version of the main theorem of 2] to arrive at a contradiction. One of the main results of 4] asserts: Theorem 2 (Bowditch 4]). If @? has a global cut point, then @? has a nontrivial, equivariant dendrite quotient D(@?) on which ? acts as a discrete convergence group. We refer to 4] for deenitions; a dendrite is a compact separable R-tree. We assume that @? has a global cut point, and we start with a graph of group decomposition of ? over two ended subgroups in which none of the vertex groups splits over a nite or two ended subgroup relative to the edge groups in it. We may assume that the action of ? on the associated tree is minimal, reduced, without inversions, and =? is compact. Since we are assuming that @? has a global cut point, such splittings of ? exist by 6] and 1].