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

## TL;DR

This paper investigates the structure of the normal closure of large powers of Dehn twists in Mapping Class Groups, providing a new presentation and employing advanced geometric group theory techniques.

## Contribution

It introduces a novel presentation for the normal closure of Dehn twists using projection complexes and rotating families, answering a question of Ivanov.

## Key findings

- The normal closure has a presentation with relators as commutators of disjoint twists.
- The approach combines projection complexes with rotating families across multiple spaces.
- Provides new insights into the algebraic structure of Dehn twists in mapping class groups.

## 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 consists 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.