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

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 $T=T_{+}(\varphi )$ is the forward limit tree of an atoroidal iwip element $\varphi \in Out(F_N)$ 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.

corrected the proof of Proposition 4.1, plus several minor fixes and updates; to appear in Journal of Group Theory

References in corpus (3)

• Geometric Intersection Number and analogues of the Curve Complex for free groups
• Intersection form, laminations and currents on free groups
• Ping-pong and Outer space

Cited by in corpus (4)

• The Poisson boundary of $\text{Out}(F_N)$
• Boundaries of relative factor graphs and subgroup classification for automorphisms of free products
• Limit trees for free group automorphisms: universality
• Loxodromic elements in the cyclic splitting complex and their centralizers