Free products from spinning and rotating families

TL;DR

This paper unifies and simplifies existing geometric approaches to understanding when the normal closure of subgroups in a group forms a free product, building on the work of Dahmani-Guirardel-Osin and others.

Contribution

It provides a unified framework that simplifies and shortens the proofs of key theorems regarding free products from spinning and rotating families.

Findings

Unified existing results into a single framework.

Simplified the proof of the Dahmani-Guirardel-Osin theorem.

Extended geometric conditions for free products.

Abstract

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.