Abelian subgroups of Out(Fn)

Feighn, Mark; Handel, Michael

doi:10.2140/gt.2009.13.1657

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

Mark Feighn and Michael Handel

Geometry & Topology 13 (2009) 1657–1727

DOI: 10.2140/gt.2009.13.1657

Abstract

We classify abelian subgroups of $O u t (F_{n})$ up to finite index in an algorithmic and computationally friendly way. A process called disintegration is used to canonically decompose a single rotationless element $ϕ$ into a composition of finitely many elements and then these elements are used to generate an abelian subgroup $A (ϕ)$ that contains $ϕ$ . 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 $O u t (F_{n})$ and of ${IA}_{n}$ .

Keywords

outer automorphism, free group, train track

Mathematical Subject Classification 2000

Primary: 20F65

Secondary: 20F28

References

Publication

Received: 21 February 2007

Revised: 25 February 2009

Accepted: 7 March 2008

Published: 5 March 2009

Proposed: Benson Farb

Seconded: Martin Bridson, Joan Birman

Authors

Mark Feighn
Math Department Rutgers University Newark, NJ 07102 USA

Michael Handel
Math Department Lehman College Bronx, NY 10468 USA

Volume 13, issue 3 (2009)