Abelian subgroups of $\mathsf{Out}(F_n)$

Abstract

We classify abelian subgroups of $Out\left(F_n\right)$ up to finite index in an algorithmic and computationally friendly way. A process called disintegration is used to canonically decompose a single rotationless element $\varphi$ into a composition of finitely many elements and then these elements are used to generate an abelian subgroup $\mathcal{A}\left(\varphi \right)$ that contains $\varphi$. The main theorem is that up to finite index every abelian subgroup is realized by this construction. As an application we give an explicit description, in terms of relative train track maps and up to finite index, of all maximal rank abelian subgroups of $Out\left(F_n\right)$ and of ${IA}_n$.