  The far-reaching work of Dahmani-Guirardel-Osin and recent work of
Clay-Mangahas-Margalit provide geometric approaches to the study of the normal
closure of a subgroup (or a collection of subgroups)in an ambient group $G$.
Their work gives conditions under which the normal closure in $G$ is a free
product. In this paper we unify their results and simplify and significantly
shorten the proof of the Dahmani-Guirardel-Osin theorem.
