On the geometry of Outer space

Vogtmann, Karen

doi:10.1090/S0273-0979-2014-01466-1

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
Full-text PDF Free Access

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 $\mathrm {Out}(F_n)$ .

[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 477161, https://doi.org/10.1007/BF02545743
[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, 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, 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 𝑂𝑢𝑡(𝐹_{𝑛})-complex, Groups Geom. Dyn. 4 (2010), no. 1, 31–58. MR 2566300, 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, 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, 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, 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, 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, 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, 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 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, 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] F. Laudenbach, Sur les 2-sphères d’une variété de dimension 3, Ann. of Math. (2) 97 (1973), 57–81 (French). MR 314054, https://doi.org/10.2307/1970877
[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, 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, 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, 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 , 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.