Reducing systems for very small trees

TL;DR

This paper investigates very small trees through reducing systems of free factors, characterizes arational trees, and applies these insights to classify elements of Out(FN) and describe the Gromov boundary of the free factor complex.

Contribution

It introduces a characterization of arational trees and provides new control over reducing systems, extending the understanding of free factor actions and their applications.

Findings

Arational trees are either free and indecomposable or dual to a surface with an arational foliation.

Provides a classification theorem for elements of Out(FN).

Establishes control over the collection of all free factors reducing a given tree.

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.