#### Volume 13, issue 3 (2009)

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

### Mark Feighn and Michael Handel

Geometry & Topology 13 (2009) 1657–1727
##### 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 $\mathsc{A}\left(\varphi \right)$ 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 ${IA}_{n}$.

##### Keywords
outer automorphism, free group, train track
Primary: 20F65
Secondary: 20F28