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

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

2009-04-12

arxiv.org/abs/0904.1881v3

Mathematics

Group Theory

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

Scientific paper

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.

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