On indecomposable trees in the boundary of Outer space

TL;DR

This paper investigates indecomposable $b R$-trees with free group actions, showing that dense orbits of finitely generated subgroups occur only when the subgroup has finite index, with applications to dual laminations.

Contribution

It establishes a characterization of subgroup actions on indecomposable trees and links this to dual lamination properties, generalizing previous results.

Findings

Subgroups with dense orbits are exactly those of finite index in $F_n$.

For indecomposable free trees, subgroups carry leaves of the dual lamination iff they are finite index.

Generalizes results on stable trees of fully irreducible automorphisms.

Abstract

Let $T$ be an $\mathbb{R}$-tree, equipped with a very small action of the rank $n$ free group $F_n$, and let $H \leq F_n$ be finitely generated. We consider the case where the action $F_n \curvearrowright T$ is indecomposable--this is a strong mixing property introduced by Guirardel. In this case, we show that the action of $H$ on its minimal invarinat subtree $T_H$ has dense orbits if and only if $H$ is finite index in $F_n$. There is an interesting application to dual algebraic laminations; we show that for $T$ free and indecomposable and for $H \leq F_n$ finitely generated, $H$ carries a leaf of the dual lamination of $T$ if and only if $H$ is finite index in $F_n$. This generalizes a result of Bestvina-Feighn-Handel regarding stable trees of fully irreducible automorphisms.