On continuous extensions of grafting maps

The definition of the grafting operation for quasifuchsian groups is extended by Bromberg (preprint) to all -groups. In this paper, we show that the extended grafting maps behave as continuous maps for every sequence which converges ``standardly' to a boundary group, although the maps are not continuous in general. As a consequence of this result, we extend Goldman's grafting theorem for quasifuchsian groups to all boundary -groups.