The normal closure of big Dehn twists and plate spinning with rotating families

Abstract

We study the normal closure of a big power of one or several Dehn twists in a mapping class group. We prove that it has a presentation whose relators consist only of commutators between twists of disjoint support, thus answering a question of Ivanov. Our method is to use the theory of projection complexes of Bestvina, Bromberg and Fujiwara, together with the theory of rotating families, simultaneously on several spaces.