Local cut points and splittings of relatively hyperbolic groups

We show that the existence of a nonparabolic local cut point in the Bowditch boundary [math] of a relatively hyperbolic group [math] implies that [math] splits over a [math] –ended subgroup. This theorem generalizes a theorem of Bowditch from the setting of hyperbolic groups to relatively hyperbolic groups. As a consequence we are able to generalize a theorem of Kapovich and Kleiner by classifying the homeomorphism type of [math] –dimensional Bowditch boundaries of relatively hyperbolic groups which satisfy certain properties, such as no splittings over [math] –ended subgroups and no peripheral splittings.¶ In order to prove the boundary classification result we require a notion of ends of a group which is more general than the standard notion. We show that if a finitely generated discrete group acts properly and cocompactly on two generalized Peano continua [math] and [math] , then [math] is homeomorphic to [math] . Thus we propose an alternative definition of [math] which increases the class of spaces on which [math] can act.