Reducing systems for very small trees
Patrick Reynolds

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 reduces if and only if acts on some subtree of with dense orbits. We characterize those trees, called arational, which do not admit a reduction by any free factor: is arational if and only if either is free and indecomposable or 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 . We also include an appendix containing examples of very small trees. The results of this paper…
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Taxonomy
TopicsGeometric and Algebraic Topology · Mathematical Dynamics and Fractals · Geometric Analysis and Curvature Flows
