no. 6
A compactification of outer space which is an absolute retract
Annales de l'Institut Fourier, Volume 69 (2019) no. 6, pp. 2395-2437.

We define a new compactification of outer space CV N (the Pacman compactification) which is an absolute retract, for which the boundary is a Z-set. The classical compactification CV N ¯ made of very small F N -actions on ℝ-trees, however, fails to be locally 4-connected as soon as N≥4. The Pacman compactification is a blow-up of CV N ¯, obtained by assigning an orientation to every arc with nontrivial stabilizer in the trees.

Nous définissons une nouvelle compactification de l’outre espace CV N (la compactification de Pacman) qui est un rétract absolu et dont le bord est un Z-ensemble. À l’inverse, pour tout N≥4, la compactification classique CV N ¯, qui consiste en les actions très petites de F N sur des arbres réels, n’est pas localement 4-connexe. La compactification de Pacman est un éclatement de CV N ¯, obtenu en attribuant une orientation à tout arc à stabilisateur non trivial dans ces arbres réels.

Received: 2016-01-12
Revised: 2017-02-13
Accepted: 2018-11-06
Published online: 2019-10-29
DOI: https://doi.org/10.5802/aif.3298
Classification: 20E36
Keywords: Outer space, compactification, absolute retract, Z-set, Out(F N ) 
@article{AIF_2019__69_6_2395_0,
     author = {Bestvina, Mladen and Horbez, Camille},
     title = {A compactification of outer space which is an absolute retract},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {6},
     year = {2019},
     pages = {2395-2437},
     doi = {10.5802/aif.3298},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2019__69_6_2395_0/}
}
Bestvina, Mladen; Horbez, Camille. A compactification of outer space which is an absolute retract. Annales de l'Institut Fourier, Volume 69 (2019) no. 6, pp. 2395-2437. doi : 10.5802/aif.3298. https://aif.centre-mersenne.org/item/AIF_2019__69_6_2395_0/

[1] Bartels, Arthur Coarse flow spaces for relatively hyperbolic groups, Compos. Math., Tome 153 (2017) no. 4, pp. 745-779 | Article | MR 3631229 | Zbl 1366.18012

[2] Bartels, Arthur; Bestvina, Mladen The Farrell–Jones Conjecture for mapping class groups, Invent. Math., Tome 215 (2019) no. 2, pp. 651-712 | Article | MR 3910072 | Zbl 07023770

[3] Bartels, Arthur; Lück, Wolfgang; Reich, Holger The K-theoretic Farrell–Jones conjecture for hyperbolic groups, Invent. Math., Tome 172 (2008) no. 1, pp. 29-70 | Article | MR 2385666 | Zbl 1143.19003

[4] Bestvina, Mladen; Feighn, Mark Outer limits (1994) (preprint, available at http://andromeda.rutgers.edu/~feighn/papers/outer.pdf)

[5] Bestvina, Mladen; Guirardel, Vincent; Horbez, Camille Boundary amenability of Out(F N ) (2017) (https://arxiv.org/abs/1705.07017)

[6] Bestvina, Mladen; Mess, Geoffrey The boundary of negatively curved groups, J. Am. Math. Soc., Tome 4 (1991) no. 3, pp. 469-481 | Article | MR 1096169 | Zbl 0767.20014

[7] Borsuk, Karol Theory of retracts, Monografie Matematyczne, Tome 44, Państwowe Wydawnictwo Naukowe, 1967, 251 pages | MR 216473 | Zbl 0153.52905

[8] Cohen, Marshall M.; Lustig, Martin Very small group actions on ℝ-trees and Dehn twist automorphisms, Topology, Tome 34 (1995) no. 3, pp. 575-617 | Article | MR 1341810 | Zbl 0844.20018

[9] Culler, Marc; Morgan, John W. Group actions on ℝ-trees, Proc. Lond. Math. Soc., Tome 55 (1987) no. 3, pp. 571-604 | Article | Zbl 0658.20021

[10] Culler, Marc; Vogtmann, Karen Moduli of graphs and automorphisms of free groups, Invent. Math., Tome 84 (1986) no. 1, pp. 91-119 | Article | MR 830040 | Zbl 0589.20022

[11] Culler, Marc; Vogtmann, Karen The boundary of outer space in rank two, Arboreal group theory (Berkeley, CA, 1988) (Mathematical Sciences Research Institute Publications) Tome 19 (1991), pp. 189-230 | Article | MR 1105335 | Zbl 0786.57002

[12] Dugundji, James Absolute neighborhood retracts and local connectedness in arbitrary metric spaces, Compos. Math., Tome 13 (1958), pp. 229-246 | MR 113217 | Zbl 0089.38903

[13] Eilenberg, Samuel; Steenrod, Norman Foundations of algebraic topology, Princeton University Press, 1952, xv+328 pages | Article | Zbl 0047.41402

[14] Engelking, Ryszard Dimension theory, North-Holland, 1978 | Zbl 0401.54029

[15] Gaboriau, Damien; Levitt, Gilbert The rank of actions on ℝ-trees, Ann. Sci. Éc. Norm. Supér., Tome 28 (1995) no. 5, pp. 549-570 | Article | MR 1341661 | Zbl 0835.20038

[16] Guirardel, Vincent Approximations of stable actions on ℝ-trees, Comment. Math. Helv., Tome 73 (1998) no. 1, pp. 89-121 | Article | MR 1610591 | Zbl 0979.20026

[17] Guirardel, Vincent; Levitt, Gilbert Deformation spaces of trees, Groups Geom. Dyn., Tome 1 (2007) no. 2, pp. 135-181 | Article | MR 2319455 | Zbl 1134.20026

[18] Guirardel, Vincent; Levitt, Gilbert The outer space of a free product, Proc. Lond. Math. Soc., Tome 94 (2007) no. 3, pp. 695-714 | Article | MR 2325317 | Zbl 1168.20011

[19] Hatcher, Allen E. Measured lamination spaces for surfaces, from the topological viewpoint, Topology Appl., Tome 30 (1988) no. 1, pp. 63-88 | Article | MR 964063 | Zbl 0662.57005

[20] Horbez, Camille Hyperbolic graphs for free products, and the Gromov boundary of the graph of cyclic splittings, J. Topol., Tome 9 (2016) no. 2, pp. 401-450 | Article | MR 3509969 | Zbl 1361.20029

[21] Horbez, Camille Spectral rigidity for primitive elements of F N , J. Group Theory, Tome 19 (2016) no. 1, pp. 55-123 | MR 3441129 | Zbl 1350.20022

[22] Horbez, Camille The boundary of the outer space of a free product, Isr. J. Math., Tome 221 (2017) no. 1, pp. 179-234 | Article | MR 3705852 | Zbl 1414.20010

[23] Hu, Sze-Tsen Theory of retracts, Wayne State University Press, 1965, 234 pages | Zbl 0145.43003

[24] Ivanov, Nikolai V. Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, Tome 115, American Mathematical Society, 1992, xii+127 pages (Translated from the Russian by E. J. F. Primrose and revised by the author) | MR 1195787

[25] Kuratowski, Kasimir Sur les espaces localement connexes et péaniens en dimension n, Fundam. Math., Tome 24 (1935) no. 1, pp. 269-287 | Article | Zbl 0011.04002

[26] Kuratowski, Kasimir Topology. Vol. II, Academic Press; Państwowe Wydawnictwo Naukowe Polish Scientific Publishers, 1968, xiv+608 pages (New edition, revised and augmented. Translated from the French by A. Kirkor)

[27] Levitt, Gilbert Graphs of actions on ℝ-trees, Comment. Math. Helv., Tome 69 (1994) no. 1, pp. 28-38 | Article | MR 1259604 | Zbl 0802.05044

[28] Levitt, Gilbert; Paulin, Frédéric Geometric group actions on trees, Am. J. Math., Tome 119 (1997) no. 1, pp. 83-102 | Article | MR 1428059 | Zbl 0878.20019

[29] Paulin, Frédéric Topologie de Gromov équivariante, structures hyperboliques et arbres réels, Invent. Math., Tome 94 (1988) no. 1, pp. 53-80 | Article | Zbl 0673.57034

[30] Paulin, Frédéric The Gromov topology on ℝ-trees, Topology Appl., Tome 32 (1989) no. 3, pp. 197-221 | Article | MR 1007101 | Zbl 0675.20033

[31] Penner, Robert C. A construction of pseudo-Anosov homeomorphisms, Trans. Am. Math. Soc., Tome 310 (1988) no. 1, pp. 179-197 | Article | MR 930079 | Zbl 0706.57008

[32] Serre, Jean-Pierre Arbres, amalgames, SL 2 , Astérisque, Tome 46, Société Mathématique de France, 1977 | Zbl 0302.20039

[33] Shenitzer, Abe Decomposition of a group with a single defining relation into a free product, Proc. Am. Math. Soc., Tome 6 (1955), pp. 273-279 | Article | MR 69174 | Zbl 0064.02204

[34] Skora, Richard K. Deformations of length functions in groups (1989) (preprint)

[35] Skora, Richard K. Splittings of surfaces, J. Am. Math. Soc., Tome 9 (1996) no. 2, pp. 605-616 | Article | MR 1339846 | Zbl 0877.57002

[36] Swarup, G. A. Decompositions of free groups, J. Pure Appl. Algebra, Tome 40 (1986) no. 1, pp. 99-102 | Article | MR 825183 | Zbl 0579.20021

[37] Thurston, William P. On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Am. Math. Soc., Tome 19 (1988) no. 2, pp. 417-431 | Article | MR 956596 | Zbl 0674.57008