Stabilizers of $\mathbb R$-trees with free isometric actions of $F_N$

TL;DR

This paper proves that the stabilizer of an $ eal$-tree with a free isometric $F_N$ action is virtually cyclic and provides a new proof of the Tits alternative for subgroups of $Out(F_N)$ containing an iwip element.

Contribution

It establishes the virtual cyclicity of stabilizers for certain $ eal$-trees and offers a novel proof of the Tits alternative for specific subgroups of $Out(F_N)$.

Findings

Stabilizers of minimal free isometric $F_N$-actions on $ eal$-trees are virtually cyclic.

A new proof of the Tits alternative for subgroups of $Out(F_N)$ containing an iwip.

The results extend understanding of subgroup structures in $Out(F_N)$.

Abstract

We prove that if $T$ is an $\mathbb R$-tree with a minimal free isometric action of $F_N$, then the $Out(F_N)$-stabilizer of the projective class $[T]$ is virtually cyclic. For the special case where $T=T_+(\phi)$ is the forward limit tree of an atoroidal iwip element $\phi\in Out(F_N)$ this is a consequence of the results of Bestvina, Feighn and Handel, via very different methods. We also derive a new proof of the Tits alternative for subgroups of $Out(F_N)$ containing an iwip (not necessarily atoroidal): we prove that every such subgroup $G\le Out(F_N)$ is either virtually cyclic or contains a free subgroup of rank two. The general case of the Tits alternative for subgroups of $Out(F_N)$ is due to Bestvina, Feighn and Handel.