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On indecomposable trees in the boundary of outer space

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Abstract

Let T be an \({\mathbb{R}}\)-tree, equipped with a very small action of the rank n free group F n , and let H ≤ 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 Guirardel. In this case, we show that the action of H on its minimal invarinat 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 ≤ 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.

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  1. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL, 61801, USA

    Patrick Reynolds

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  1. Patrick Reynolds
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Correspondence to Patrick Reynolds.

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Reynolds, P. On indecomposable trees in the boundary of outer space. Geom Dedicata 153, 59–71 (2011). https://doi.org/10.1007/s10711-010-9556-x

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