Reducing systems for very small trees

Abstract

We study very small trees from the point of view of reducing systems of free factors, which are analogues of reducing systems of curves for a surface lamination; a non-trivial, proper free factor $F \leq \FN$ reduces $T$ if and only if $F$ acts on some subtree of $T$ with dense orbits. We characterize those trees, called arational, which do not admit a reduction by any free factor: $T$ is arational if and only if either $T$ is free and indecomposable or $T$ is dual to a surface with one boundary component equipped with an arational measured foliation. To complement this result, we establish some results giving control over the collection of all factors reducing a given tree. As an application, we deduce a form of the celebrated Bestvina-Handel classification theorem for elements of $Out(\FN)$. We also include an appendix containing examples of very small trees. The results of this paper are used in Bestvina and Reynolds (2012), where we describe the Gromov boundary of the complex of free factors.

32 pages

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• Boundaries of relative factor graphs and subgroup classification for automorphisms of free products
• Ergodic decompositions for folding and unfolding paths in Outer space
• A short proof of Handel and Mosher's alternative for subgroups of $\text{Out}(F_N)$
• Random trees in the boundary of Outer space
• Measure equivalence rigidity of $\mathrm{Out}(F_N)$
• Constructing non-uniquely ergodic arational trees
• Loxodromic elements in the cyclic splitting complex and their centralizers
• Cocycle superrigidity from higher rank lattices to $\mathrm{Out}(F_N)$