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On the geometry of Outer space


Author: Karen Vogtmann
Journal: Bull. Amer. Math. Soc. 52 (2015), 27-46
MSC (2010): Primary 20F65
DOI: https://doi.org/10.1090/S0273-0979-2014-01466-1
Published electronically: August 19, 2014
MathSciNet review: 3286480
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Abstract | References | Similar Articles | Additional Information

Abstract: Outer space is a space of graphs used to study the group $ \mathrm {Out}(F_n)$ of outer automorphisms of a finitely generated free group. We discuss an emerging metric theory for Outer space and some applications to $ \mathrm {Out}(F_n)$.


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  • [1] Yael Algom-Kfir, Strongly contracting geodesics in outer space, Geom. Topol. 15 (2011), no. 4, 2181-2233. MR 2862155, https://doi.org/10.2140/gt.2011.15.2181
  • [2] Yael Algom-Kfir, The Metric Completion of Outer Space, arXiv:1202.6392.
  • [3] Yael Algom-Kfir and Mladen Bestvina, Asymmetry of outer space, Geom. Dedicata 156 (2012), 81-92. MR 2863547, https://doi.org/10.1007/s10711-011-9591-2
  • [4] Lipman Bers, An extremal problem for quasiconformal mappings and a theorem by Thurston, Acta Math. 141 (1978), no. 1-2, 73-98. MR 0477161 (57 #16704)
  • [5] Gregory C. Bell and Koji Fujiwara, The asymptotic dimension of a curve graph is finite, J. Lond. Math. Soc. (2) 77 (2008), no. 1, 33-50. MR 2389915 (2009d:57037), https://doi.org/10.1112/jlms/jdm090
  • [6] Mladen Bestvina, A Bers-like proof of the existence of train tracks for free group automorphisms, Fund. Math. 214 (2011), no. 1, 1-12. MR 2845630 (2012m:20046), https://doi.org/10.4064/fm214-1-1
  • [7] Mladen Bestvina, PCMI Lectures on the geometry of Outer space, (2014), 1-34.
  • [8] Mladen Bestvina, Kenneth Bromberg, and Koji Fujiwara. Constructing group actions on quasi-trees and applications to mapping class groups, arXiv:1006.1939.
  • [9] Mladen Bestvina and Mark Feighn, A hyperbolic $ {\rm Out}(F_n)$-complex, Groups Geom. Dyn. 4 (2010), no. 1, 31-58. MR 2566300 (2011a:20052), https://doi.org/10.4171/GGD/74
  • [10] Mladen Bestvina and Mark Feighn, Hyperbolicity of the complex of free factors, arXiv:1107.3308. (2011).
  • [11] Mladen Bestvina and Mark Feighn, Subfactor projections, arXiv:1211.1730.
  • [12] Mladen Bestvina and Koji Fujiwara, Quasi-homomorphisms on mapping class groups, Glas. Mat. Ser. III 42(62) (2007), no. 1, 213-236. MR 2332668 (2008k:57002), https://doi.org/10.3336/gm.42.1.15
  • [13] Mladen Bestvina and Michael Handel, Train tracks and automorphisms of free groups, Ann. of Math. (2) 135 (1992), no. 1, 1-51. MR 1147956 (92m:20017), https://doi.org/10.2307/2946562
  • [14] Brian H. Bowditch, Intersection numbers and the hyperbolicity of the curve complex, J. Reine Angew. Math. 598 (2006), 105-129. MR 2270568 (2009b:57034), https://doi.org/10.1515/CRELLE.2006.070
  • [15] Daryl Cooper, Automorphisms of free groups have finitely generated fixed point sets, J. Algebra 111 (1987), no. 2, 453-456. MR 916179 (89a:20024), https://doi.org/10.1016/0021-8693(87)90229-8
  • [16] Marc Culler and Karen Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), no. 1, 91-119. MR 830040 (87f:20048), https://doi.org/10.1007/BF01388734
  • [17] Stefano Francaviglia and Armando Martino, Metric properties of Outer space, Publ. Mat. 55 (2011), no. 2, 433-473. MR 2839451 (2012j:20128), https://doi.org/10.5565/PUBLMAT_55211_09
  • [18] Ursula Hamenstädt, Lines of minima in Outer space, Duke Math. J. 163 (2014), no. 4, 733-776. MR 3178431, https://doi.org/10.1215/00127094-2429807
  • [19] Ursula Hamenstädt and Sebastian Hensel, Spheres and Projections for $ \mathrm {Out}(F_n)$, arXiv:1109.2687.
  • [20] Michael Handel and Lee Mosher, Axes in Outer space, Mem. Amer. Math. Soc. 213 (2011), no. 1004, vi+104. MR 2858636, https://doi.org/10.1090/S0065-9266-2011-00620-9
  • [21] Michael Handel and Lee Mosher, Lipschitz retraction and distortion for subgroups of $ {\rm Out}(F_n)$, Geom. Topol. 17 (2013), no. 3, 1535-1579. MR 3073930, https://doi.org/10.2140/gt.2013.17.1535
  • [22] Michael Handel and Lee Mosher, The free splitting complex of a free group, I: hyperbolicity, Geom. Topol. 17 (2013), no. 3, 1581-1672. MR 3073931, https://doi.org/10.2140/gt.2013.17.1581
  • [23] Michael Handel and Lee Mosher, Subgroup decomposition in $ \mathrm {Out}(F_n)$, Part I: Geometric Models, arXiv:1302.2378.
  • [24] Michael Handel and Lee Mosher, Subgroup decomposition in $ \mathrm {Out}(F_n)$, Part II: A relative Kolchin theorem, arXiv:1302.2379.
  • [25] Michael Handel and Lee Mosher, Subgroup decomposition in $ \mathrm {Out}(F_n)$, Part III: Weak attraction theory, arXiv:1306.4712.
  • [26] Michael Handel and Lee Mosher, Subgroup decomposition in $ \mathrm {Out}(F_n)$, Part IV: Relatively irreducible subgroups, arXiv:1306.4711.
  • [27] Allen Hatcher, Homological stability for automorphism groups of free groups, Comment. Math. Helv. 70 (1995), no. 1, 39-62. MR 1314940 (95k:20030), https://doi.org/10.1007/BF02565999
  • [28] Arnaud Hilion and Camille Horbez, The hyperbolicity of the sphere complex via surgery paths, arXiv:1210.6183.
  • [29] Camille Horbez, Sphere paths in outer space, Algebr. Geom. Topol. 12 (2012), no. 4, 2493-2517. MR 3020214, https://doi.org/10.2140/agt.2012.12.2493
  • [30] Ilya Kapovich and Kasra Rafi, On hyperbolicity of free splitting and free factor complexes, arXiv:1206.3626.
  • [31] Francois Laudenbach, Sur les $ 2$-spheres d'une varièté de dimension $ 3$, Ann. of Math. 97 (1973) 57-81. MR 0314054
  • [32] Howard A. Masur and Yair N. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999), no. 1, 103-149. MR 1714338 (2000i:57027), https://doi.org/10.1007/s002220050343
  • [33] H. A. Masur and Y. N. Minsky, Geometry of the complex of curves. II. Hierarchical structure, Geom. Funct. Anal. 10 (2000), no. 4, 902-974. MR 1791145 (2001k:57020), https://doi.org/10.1007/PL00001643
  • [34] Yair N. Minsky, Quasi-projections in Teichmüller space, J. Reine Angew. Math. 473 (1996), 121-136. MR 1390685 (97b:32020), https://doi.org/10.1515/crll.1995.473.121
  • [35] Lucas Sabalka and Dmitri Savchuk, Submanifold projection, arXiv:1211.3111.
  • [36] Lucas Sabalka and Dmitri Savchuk, On the geometry of a proposed curve complex analogue for $ Out(F_n)$, arXiv:1007.1998.
  • [37] John R. Stallings, Finite graphs and free groups, Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982) Contemp. Math., vol. 44, Amer. Math. Soc., Providence, RI, 1985, pp. 79-84. MR 813103, https://doi.org/10.1090/conm/044/813103
  • [38] Samuel J. Taylor, A note on subfactor projections, Algebr. Geom. Topol. 14 (2014), no. 2, 805-821. MR 3159971, https://doi.org/10.2140/agt.2014.14.805
  • [39] William P. Thurston, The geometry and topology of three-manifolds, Princeton Univ. Press (1978).
  • [40] William P. Thurston, Minimal stretch maps between hyperbolic surfaces, arXiv:9801.039.

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Additional Information

Karen Vogtmann
Affiliation: University of Warwick and Cornell University
Email: k.vogtmann@warwick.ac.uk

DOI: https://doi.org/10.1090/S0273-0979-2014-01466-1
Received by editor(s): May 31, 2014
Published electronically: August 19, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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