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Axes in outer space. (English) Zbl 1238.57002
Mem. Am. Math. Soc. 1004, v, 104 p. (2011).
The authors develop a notion of axis-bundle for the action of a nongeometric, fully irreducible outer automorphism on outer space. The Culler-Vogtmann outer space $$X_r$$ of a finite rank free group $$F_r$$ has no natural metric. Hence there does not seem to be a natural notion of an axis for a fully irreducible $$\phi \in Out(F_r)$$. Nevertheless, it is shown in the paper under review that the axes for $$\phi$$ fit into an axis bundle $$A_\phi$$ satisfying the following topological properties:
1.
$$A_\phi$$ is a closed subset of $$X_r$$ proper homotopy equivalent to a line,
2.
$$A_\phi$$ is invariant under $$\phi$$,
3.
The two boundary points of $$A_\phi$$ on the boundary of outer space correspond to the repeller and attractor of the source-sink action of $$\phi$$, i.e. the negative end of $$A_\phi$$ limits in compactified outer space on the repeller, and the positive end limits on the attractor.
4.
$$A_\phi$$ depends naturally on the repeller and attractor, i.e. for any nongeometric, fully irreducible $$\phi,\phi' \in Out(F_r)$$ and any $$\psi \in Out(F_r)$$, if $$\psi$$ takes the repeller-attractor pair of $$\phi$$ to the repeller-attractor pair of $$\phi'$$, then $$\psi$$ takes $$A_\phi$$ to $$A_{\phi'}$$. It follows that $$A_\phi$$ is invariant under $$\phi$$ and that $$\phi^n$$ and $$\phi$$ have the same axis bundle for all $$n>0$$.
The authors propose various definitions for $$A_\phi$$, motivated by various aspects of the theory of outer automorphisms, and by analogy with various properties of Teichmüller geodesics. The main thrust of the paper is to prove the equivalence between these definitions.
Reviewer: Mahan Mj (Howrah)

##### MSC:
 57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes 20-02 Research exposition (monographs, survey articles) pertaining to group theory 20F65 Geometric group theory 57M07 Topological methods in group theory
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##### References:
 [1] Mladen Bestvina and Noel Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), no. 3, 445-470. · Zbl 0888.20021 · doi:10.1007/s002220050168 [2] M. Bestvina and M. Feighn, Outer limits, preprint. [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 [4] Mladen Bestvina, Mark Feighn, and Michael Handel, The Tits alternative for $${\mathrm Out}(F_n)$$. I. Dynamics of exponentially-growing automorphisms, Ann. of Math. (2) 151 (2000), no. 2, 517-623. · Zbl 0984.20025 · doi:10.2307/121043 [5] Mladen Bestvina, Mark Feighn, and Michael Handel, Solvable subgroups of $${\mathrm Out}(F_n)$$ are virtually Abelian, Geom. Dedicata 104 (2004), 71-96. · Zbl 1052.20027 · doi:10.1023/B:GEOM.0000022864.30278.34 [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 [7] Matt Clay, Contractibility of deformation spaces of $$G$$-trees, Algebr. Geom. Topol. 5 (2005), 1481-1503 (electronic). · Zbl 1120.20027 · doi:10.2140/agt.2005.5.1481 [8] Marc Culler and John W. Morgan, Group actions on $${\mathbf R}$$-trees, Proc. London Math. Soc. (3) 55 (1987), no. 3, 571-604. · Zbl 0658.20021 · doi:10.1112/plms/s3-55.3.571 [9] Daryl Cooper, Automorphisms of free groups have finitely generated fixed point sets, J. Algebra 111 (1987), no. 2, 453-456. · Zbl 0628.20029 · doi:10.1016/0021-8693(87)90229-8 [10] M. Coornaert and A. Papadopoulos, Une dichotomie de Hopf pour les flots géodésiques associés aux groupes discrets d’isométries des arbres, Trans. Amer. Math. Soc. 343 (1994), no. 2, 883-898 (French). · Zbl 0816.58033 · doi:10.1090/S0002-9947-1994-1207579-2 [11] 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 [12] Mark Feighn and Michael Handel, Abelian subgroups of $${\mathrm Out}(F_n)$$, Geom. Topol. 13 (2009), no. 3, 1657-1727. · Zbl 1201.20031 · doi:10.2140/gt.2009.13.1657 [13] 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 [14] Étienne Ghys and Pierre de la Harpe, Infinite groups as geometric objects (after Gromov), Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989), Oxford Sci. Publ., Oxford Univ. Press, New York, 1991, pp. 299-314. · Zbl 0764.57003 [15] 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 [16] Vincent Guirardel and Gilbert Levitt, The outer space of a free product, Proc. Lond. Math. Soc. (3) 94 (2007), no. 3, 695-714. · Zbl 1168.20011 · doi:10.1112/plms/pdl026 [17] S. M. Gersten and J. R. Stallings, Irreducible outer automorphisms of a free group, Proc. Amer. Math. Soc. 111 (1991), no. 2, 309-314. · Zbl 0717.20026 · doi:10.1090/S0002-9939-1991-1052571-7 [18] Michael Handel and Lee Mosher, Parageometric outer automorphisms of free groups, Trans. Amer. Math. Soc. 359 (2007), no. 7, 3153-3183 (electronic). · Zbl 1120.20042 · doi:10.1090/S0002-9947-07-04065-2 [19] Michael Handel and Lee Mosher, The expansion factors of an outer automorphism and its inverse, Trans. Amer. Math. Soc. 359 (2007), no. 7, 3185-3208 (electronic). · Zbl 1127.20021 · doi:10.1090/S0002-9947-07-04066-4 [20] Michael Handel and William P. Thurston, New proofs of some results of Nielsen, Adv. in Math. 56 (1985), no. 2, 173-191. · Zbl 0584.57007 · doi:10.1016/0001-8708(85)90028-3 [21] Gilbert Levitt, Graphs of actions on $${\mathbf R}$$-trees, Comment. Math. Helv. 69 (1994), no. 1, 28-38. · Zbl 0802.05044 · doi:10.1007/BF02564472 [22] Gilbert Levitt and Martin Lustig, Periodic ends, growth rates, Hölder dynamics for automorphisms of free groups, Comment. Math. Helv. 75 (2000), no. 3, 415-429. · Zbl 0965.20026 · doi:10.1007/s000140050134 [23] Gilbert Levitt and Martin Lustig, Irreducible automorphisms of $$F_n$$ 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 [24] J. Los and M. Lustig, The set of train track representatives of an irreducible free group automorphism is contractible, preprint, December 2004. [25] M. Lustig, Structure and conjugacy for automorphisms of free groups, preprint, 1999. · Zbl 0956.20021 [26] 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 · doi:10.1090/S0002-9947-1985-0800253-8 [27] Howard Masur and John Smillie, Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms, Comment. Math. Helv. 68 (1993), no. 2, 289-307. · Zbl 0792.30030 · doi:10.1007/BF02565820 [28] Jakob Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen, Acta Math. 50 (1927), no. 1, 189-358 (German). · JFM 53.0545.12 · doi:10.1007/BF02421324 [29] Frédéric Paulin, The Gromov topology on $${\mathbf R}$$-trees, Topology Appl. 32 (1989), no. 3, 197-221. · Zbl 0675.20033 · doi:10.1016/0166-8641(89)90029-1 [30] R. Skora, Deformations of length functions in groups, preprint. · Zbl 0607.57008 [31] Richard K. Skora, Splittings of surfaces, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 1, 85-90. · Zbl 0708.30044 · doi:10.1090/S0273-0979-1990-15907-5 [32] Edwin H. Spanier, Algebraic topology, Springer-Verlag, New York-Berlin, 1981. Corrected reprint. · Zbl 0145.43303 [33] John R. Stallings, Topologically unrealizable automorphisms of free groups, Proc. Amer. Math. Soc. 84 (1982), no. 1, 21-24. · Zbl 0477.20012 · doi:10.1090/S0002-9939-1982-0633269-8 [34] John R. Stallings, Topology of finite graphs, Invent. Math. 71 (1983), no. 3, 551-565. · Zbl 0521.20013 · doi:10.1007/BF02095993 [35] 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 [36] Tad White, Fixed points of finite groups of free group automorphisms, Proc. Amer. Math. Soc. 118 (1993), no. 3, 681-688. · Zbl 0798.20021 · doi:10.1090/S0002-9939-1993-1164152-7
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