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Stabilizers of \(\mathbb{R}\)-trees with free isometric actions of \(F_N\). (English) Zbl 1262.20031
Summary: We prove that if \(T\) is an \(\mathbb{R}\)-tree with a minimal free isometric action of \(F_N\), then the \(\mathrm{Out}(F_N)\)-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 \(\varphi\in\mathrm{Out}(F_N)\) this is a consequence of the results of M. Bestvina, M. Feighn and M. Handel [Geom. Funct. Anal. 7, No. 2, 215-244 (1997; Zbl 0884.57002)], via very different methods.
We also derive a new proof of the Tits alternative for subgroups of \(\mathrm{Out}(F_N)\) containing an iwip (not necessarily atoroidal): we prove that every such subgroup \(G\leq\mathrm{Out}(F_N)\) is either virtually cyclic or contains a free subgroup of rank two. The general case of the Tits alternative for subgroups of \(\mathrm{Out}(F_N)\) is due to Bestvina, Feighn and Handel.

MSC:
20E08 Groups acting on trees
20E36 Automorphisms of infinite groups
20E05 Free nonabelian groups
20F28 Automorphism groups of groups
20E07 Subgroup theorems; subgroup growth
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