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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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References:
[1] Bestvina, M., Feighn, M.: Stable actions of groups on real trees. Invent. Math · Zbl 0837.20047
[2] Bestvina, M., Feighn, M.: Outer limits (preprint), http://andromeda.rutgers.edu/\(\sim\)feighn/papers/outer.pdf (1994)
[3] Bestvina M., Feighn M., Handel M.: Laminations, trees, and irreducible automorphisms of free groups. Geom. Funct. Anal. 7(2), 215–244 (1997) · Zbl 0884.57002 · doi:10.1007/PL00001618
[4] Bestvina M., Handel M.: Train tracks and automorphisms of free groups. Ann. Math. (2) 135(1), 1–51 (1992) · Zbl 0757.57004 · doi:10.2307/2946562
[5] Casson A.J., Bleiler S.A.: Automorphisms of Surfaces after Nielsen and Thurston, London Mathematical Society Student Texts, Vol. 9. Cambridge University Press, Cambridge (1988) · Zbl 0649.57008
[6] Chiswell I.: Introduction to \(\Lambda\)-Trees. World Scientific Publishing Company Co. Pte. Ltd., Singapore (2001) · Zbl 1004.20014
[7] Cohen M., Lustig M.: Very small group actions on $${\(\backslash\)mathbb{R}}$$ -trees and Dehn twist automorphisms. Topology 34, 575–617 (1995) · Zbl 0844.20018 · doi:10.1016/0040-9383(94)00038-M
[8] Coulbois T., Hilion A., Lustig M.: $${\(\backslash\)mathbb{R}}$$ -trees and laminations for free groups. I. Algebraic laminations. J. Lond. Math. Soc. (2) 78(3), 723–736 (2008) · Zbl 1197.20019 · doi:10.1112/jlms/jdn052
[9] Coulbois T., Hilion A., Lustig M.: $${\(\backslash\)mathbb{R}}$$ -trees and laminations for free groups. II. The dual lamination of an $${\(\backslash\)mathbb{R}}$$ -tree. J. Lond. Math. Soc. (2) 78(3), 737–754 (2008) · Zbl 1198.20023 · doi:10.1112/jlms/jdn053
[10] Culler M., Vogtmann K.: Moduli of graphs and automorphisms of free groups. Invent. Math. 84(1), 91–119 (1986) · Zbl 0589.20022 · doi:10.1007/BF01388734
[11] Gaboriau D., Levitt G., Paulin F.: Pseudogroups of isometries of $${\(\backslash\)mathbb{R}}$$ and Rips’ theorem on free actions on $${\(\backslash\)mathbb{R}}$$ -trees. Israel J. Math. 87, 403–428 (1994) · Zbl 0824.57001 · doi:10.1007/BF02773004
[12] Guirardel V.: Dynamics of Out(F n ) on the boundary of outer space. Ann. Sci. École Norm. Sup. 33(4), 433–465 (2000) · Zbl 1045.20034
[13] Guirardel V.: Actions of finitely generated groups on $${\(\backslash\)mathbb{R}}$$ -trees. Ann. Inst. Fourier (Grenoble) 58, 159–211 (2008) · Zbl 1187.20020 · doi:10.5802/aif.2348
[14] Hall M.: A topology for free groups and related groups. Ann. Math. (2) 52, 127–139 (1950) · Zbl 0045.31204 · doi:10.2307/1969513
[15] Kapovich I., Myasnikov A.: Stallings foldings and subgroups of free groups. J. Algebra 248(2), 608–668 (2002) · Zbl 1001.20015 · doi:10.1006/jabr.2001.9033
[16] Kapovich M.: Hyperbolic Manifolds and Discrete Groups, Progress in Mathematics, vol. 183. Birkhäuser, Boston (2001) · Zbl 0958.57001
[17] Levitt G.: Graphs of actions on $${\(\backslash\)mathbb{R}}$$ -trees. Comment. Math. Helv. 69(1), 28–38 (1994) · Zbl 0802.05044 · doi:10.1007/BF02564472
[18] Levitt G., Lustig M.: Irreducible automorphisms of F n have north-south dynamics on compactified outer space. J. Inst. Math. Jussieu 2 1, 59–72 (2003) · Zbl 1034.20038
[19] Martin R.: Non-uniquely ergodic foliations of thin type. Ergodic Theory Dynam Syst. 17(3), 667–674 (1997) · Zbl 0890.57041 · doi:10.1017/S0143385797079169
[20] Miasnikov A., Ventura E., Weil P.: Algebraic extensions in free groups. Trends Math. Birkhäuser, Basel (2007) · Zbl 1160.20022
[21] Reynolds, P.: Dynamics of irreducible endomorphisms of F n (preprint). arXiv:1008.3659 (2010)
[22] Scott P.: Subgroups of surface groups are almost geometric. J. London Math. Soc. (2) 17(3), 555–565 (1978) · Zbl 0412.57006 · doi:10.1112/jlms/s2-17.3.555
[23] Skora R.: Splittings of surfaces. J. Am. Math. Soc. 9, 605–616 (1996) · Zbl 0877.57002 · doi:10.1090/S0894-0347-96-00205-6
[24] Stallings J.R.: Topology of finite graphs. Invent. Math. 71(3), 551–565 (1983) · Zbl 0521.20013 · doi:10.1007/BF02095993
[25] Vogtmann K.: Automorphisms of free groups and outer space. Geometriae Dedicata 94, 1–31 (2002) · Zbl 1017.20035 · doi:10.1023/A:1020973910646
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