Free products from spinning and rotating families

Abstract

The far-reaching work of Dahmani–Guirardel–Osin (2017) and recent work of Clay– Mangahas–Margalit (2021) 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 theorem of Dahmani–Guirardel–Osin (2017).