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Parageometric outer automorphisms of free groups. (English) Zbl 1120.20042
The ‘expansion factor’ of an (outer) automorphism \(\varphi\) of a finitely generated group \(G\) is defined as \[ \lambda(\varphi)=\sup_{c\in \mathcal C}(\limsup_{n\to+\infty}\|\varphi^n(c)\|^{1/n}), \] where \(\mathcal C\) denotes the set of conjugacy classes of \(G\) and \(\| c \|\) the smallest word length of a representative of \(c\in\mathcal C\). If \(\varphi\) is a pseudo-Anosov element in the mapping class group of a surface (the outer automorphism group of its fundamental group) then there is a unique \(\varphi\)-invariant geodesic in Teichmüller space consisting of the points in Teichmüller space which minimize the translation distance under \(\varphi\), and this translation distance equals \(\log\lambda(\varphi)\). In the present paper, the question is considered whether this has an analogue for a fully irreducible outer automorphism \(\varphi\) of a free group \(F_r\).
The main result of the paper implies that the answer is no for ‘parageometric’ fully irreducible outer automorphisms \(\varphi\in\text{Out}(F_r)\), by showing that \(\lambda(\varphi)>\lambda(\varphi^{-1})\) for parageometric automorphisms. Here an outer automorphism \(\varphi\) is parageometric if the attracting fixed point of \(\varphi\) in the boundary of outer space is a geometric \(\mathbb{R}\)-tree with respect to the action of \(F_r\), but that \(\varphi\) itself is not geometric in that it is not represented by a homeomorphism of a bounded surface. As corollaries (proved independently by V. Guirardel, [Ann. Sci. Éc. Norm. Supér. (4) 38, No. 6, 847-888 (2005; Zbl 1110.20019)]), the inverse of a parageometric outer automorphism is neither geometric nor parageometric, and a fully irreducible outer automorphism is geometric if and only if its attracting and repelling fixed points in the boundary of outer space are geometric \(\mathbb{R}\)-trees.

MSC:
20F65 Geometric group theory
20E05 Free nonabelian groups
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
20E36 Automorphisms of infinite groups
57M07 Topological methods in group theory
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
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References:
[1] M. Bestvina and M. Feighn, Outer limits, preprint.
[2] Mladen Bestvina and Mark Feighn, Stable actions of groups on real trees, Invent. Math. 121 (1995), no. 2, 287 – 321. · Zbl 0837.20047 · doi:10.1007/BF01884300 · doi.org
[3] M. Bestvina, M. Feighn, and M. Handel, Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal. 7 (1997), no. 2, 215 – 244. · Zbl 0884.57002 · doi:10.1007/PL00001618 · doi.org
[4] Mladen Bestvina, Mark Feighn, and Michael Handel, The Tits alternative for \?\?\?(\?_\?). I. Dynamics of exponentially-growing automorphisms, Ann. of Math. (2) 151 (2000), no. 2, 517 – 623. · Zbl 0984.20025 · doi:10.2307/121043 · doi.org
[5] Mladen Bestvina, Mark Feighn, and Michael Handel, The Tits alternative for \?\?\?(\?_\?). II. A Kolchin type theorem, Ann. of Math. (2) 161 (2005), no. 1, 1 – 59. · Zbl 1139.20026 · doi:10.4007/annals.2005.161.1 · doi.org
[6] Mladen Bestvina and Michael Handel, Train tracks and automorphisms of free groups, Ann. of Math. (2) 135 (1992), no. 1, 1 – 51. · Zbl 0757.57004 · doi:10.2307/2946562 · doi.org
[7] Marshall M. Cohen and Martin Lustig, Very small group actions on \?-trees and Dehn twist automorphisms, Topology 34 (1995), no. 3, 575 – 617. · Zbl 0844.20018 · doi:10.1016/0040-9383(94)00038-M · doi.org
[8] Marc Culler and John W. Morgan, Group actions on \?-trees, Proc. London Math. Soc. (3) 55 (1987), no. 3, 571 – 604. · Zbl 0658.20021 · doi:10.1112/plms/s3-55.3.571 · doi.org
[9] Marc Culler and Karen Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), no. 1, 91 – 119. · Zbl 0589.20022 · doi:10.1007/BF01388734 · doi.org
[10] Travaux de Thurston sur les surfaces, Astérisque, vol. 66, Société Mathématique de France, Paris, 1979 (French). Séminaire Orsay; With an English summary. · Zbl 0731.57001
[11] Damien Gaboriau, Andre Jaeger, Gilbert Levitt, and Martin Lustig, An index for counting fixed points of automorphisms of free groups, Duke Math. J. 93 (1998), no. 3, 425 – 452. · Zbl 0946.20010 · doi:10.1215/S0012-7094-98-09314-0 · doi.org
[12] Vincent Guirardel, Cœur et nombre d’intersection pour les actions de groupes sur les arbres, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 6, 847 – 888 (French, with English and French summaries). · Zbl 1110.20019 · doi:10.1016/j.ansens.2005.11.001 · doi.org
[13] M. Handel and L. Mosher, The expansion factors of an outer automorphism and its inverse, Trans. Amer. Math. Soc., this issue. · Zbl 1127.20021
[14] M. Handel and L. Mosher, Axes in outer space, Preprint, arXiv:math.GR/0605355, 2006. · Zbl 1238.57002
[15] Gilbert Levitt and Martin Lustig, Irreducible automorphisms of \?_\? have north-south dynamics on compactified outer space, J. Inst. Math. Jussieu 2 (2003), no. 1, 59 – 72. · Zbl 1034.20038 · doi:10.1017/S1474748003000033 · doi.org
[16] Douglas Lind and Brian Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995. · Zbl 1106.37301
[17] Gilbert Levitt and Frédéric Paulin, Geometric group actions on trees, Amer. J. Math. 119 (1997), no. 1, 83 – 102. · Zbl 0878.20019
[18] John McCarthy, A ”Tits-alternative” for subgroups of surface mapping class groups, Trans. Amer. Math. Soc. 291 (1985), no. 2, 583 – 612. · Zbl 0579.57006
[19] Karen Vogtmann, Automorphisms of free groups and outer space, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), 2002, pp. 1 – 31. · Zbl 1017.20035 · doi:10.1023/A:1020973910646 · doi.org
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