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Abelian subgroups of \(\text{Out}(F_n)\). (English) Zbl 1201.20031
In this long paper the authors classify up to finite index the Abelian subgroups of \(\text{Out}(F_n)\), the group of outer automorphisms of the free group \(F_n\) of rank \(n\). First they construct an Abelian subgroup \(\mathcal D(\varphi)\) from a given \(\varphi\in\text{Out}(F_n)\) by a process called disintegration.
Then they prove the following (main) Theorem: For every Abelian subgroup \(A\) of \(\text{Out}(F_n)\) there exists \(\varphi\in A\) such that \(A\cap\mathcal D(\varphi)\) has finite index in \(A\).
By applying this theorem they give an explicit description, in terms of relative train track maps and up to finite index, of all maximal rank Abelian subgroups of \(\text{Out}(F_n)\) and \(\text{IA}_n\).

20F28 Automorphism groups of groups
20E07 Subgroup theorems; subgroup growth
20E05 Free nonabelian groups
20F05 Generators, relations, and presentations of groups
Full Text: DOI arXiv
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