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On indecomposable trees in the boundary of outer space. (English) Zbl 1245.20029
Summary: 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 V. Guirardel [Ann. Inst. Fourier 58, No. 1, 159-211 (2008; Zbl 1187.20020)]. In this case, we show that the action of $$H$$ on its minimal invariant 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.

##### MSC:
 20E08 Groups acting on trees 20E05 Free nonabelian groups 20F65 Geometric group theory 37B10 Symbolic dynamics 57M07 Topological methods in group theory 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
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