The horoboundary of outer space, and growth under random automorphisms

Horbez, Camille

doi:10.24033/asens.2304

The horoboundary of outer space, and growth under random automorphisms
[L'horofrontière de l'outre-espace et la croissance sous l'action d'automorphismes aléatoires]

Horbez, Camille

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 5, pp. 1075-1123.

Résumé
Abstract

Nous montrons que l'horofrontière de l'outre-espace pour la distance de Lipschitz est un quotient de la frontière classique de Culler et Morgan, dans laquelle deux arbres sont identifiés lorsque leurs fonctions-longueurs de translation sont homothétiques en restriction aux éléments primitifs de $F_{N}$ . Nous identifions l'ensemble des points de Busemann à l'ensemble des arbres à orbites denses. Nous étudions également quelques propriétés de l'horofrontière de l'outre-espace pour la distance de Lipschitz inversée, et montrons en particulier que celle-ci est de dimension topologique infinie dès que $N \geq 3$ . Nous utilisons ensuite notre description de l'horofrontière de l'outre-espace pour montrer un analogue d'un théorème de Furstenberg et Kifer [20] et Hennion [32] pour les produits aléatoires d'automorphismes extérieurs de $F_{N}$ , estimant les taux de croissance possibles des classes de conjugaison d'éléments de $F_{N}$ sous l'action de tels produits.

We show that the horoboundary of outer space for the Lipschitz metric is a quotient of Culler and Morgan's classical boundary, two trees being identified whenever their translation length functions are homothetic in restriction to the set of primitive elements of $F_{N}$ . We identify the set of Busemann points with the set of trees with dense orbits. We also investigate a few properties of the horoboundary of outer space for the backward Lipschitz metric, and show in particular that it is infinite-dimensional when $N \geq 3$ . We then use our description of the horoboundary of outer space to derive an analogue of a theorem of Furstenberg and Kifer [20] and Hennion [32] for random products of outer automorphisms of $F_{N}$ , that estimates possible growth rates of conjugacy classes of elements of $F_{N}$ under such products.

Publié le : 2016-11-12

DOI : 10.24033/asens.2304

Classification : 20F65, 20E08, 20E36, 60B15.
Keywords: $\mathrm {Out}(F_N)$, Outer space, horoboundary, growth, random walks.
Mot clés : $\mathrm {Out}(F_N)$, Outre espace, horofrontière, croissance, marches aléatoires.

@article{ASENS_2016__49_5_1075_0,
     author = {Horbez, Camille},
     title = {The horoboundary of outer space, and growth under random automorphisms},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1075--1123},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 49},
     number = {5},
     year = {2016},
     doi = {10.24033/asens.2304},
     mrnumber = {3581811},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2304/}
}

TY  - JOUR
AU  - Horbez, Camille
TI  - The horoboundary of outer space, and growth under random automorphisms
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2016
SP  - 1075
EP  - 1123
VL  - 49
IS  - 5
PB  - Société Mathématique de France. Tous droits réservés
UR  - http://www.numdam.org/articles/10.24033/asens.2304/
DO  - 10.24033/asens.2304
LA  - en
ID  - ASENS_2016__49_5_1075_0
ER  -

%0 Journal Article
%A Horbez, Camille
%T The horoboundary of outer space, and growth under random automorphisms
%J Annales scientifiques de l'École Normale Supérieure
%D 2016
%P 1075-1123
%V 49
%N 5
%I Société Mathématique de France. Tous droits réservés
%U http://www.numdam.org/articles/10.24033/asens.2304/
%R 10.24033/asens.2304
%G en
%F ASENS_2016__49_5_1075_0

Horbez, Camille. The horoboundary of outer space, and growth under random automorphisms. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 5, pp. 1075-1123. doi : 10.24033/asens.2304. http://www.numdam.org/articles/10.24033/asens.2304/

Bibliographie
Cité par

Algom-Kfir, Y. The Metric Completion of Outer Space (preprint arXiv:1202.6392 ) | MR

Algom-Kfir, Y. Strongly contracting geodesics in Outer Space, Geom. Topol., Volume 15 (2011), pp. 2181-2233 | DOI | MR | Zbl

Ballmann, W. On the Dirichlet problem at infinity for manifolds of nonpositive curvature, Forum Math., Volume 1 (1989), pp. 201-213 | DOI | MR | Zbl

Ballmann, W., DMV Seminars, 25, Birkhäuser, 1995 | MR | Zbl

Bestvina, M.; Feighn, M. Outer limits (1994) (preprint http://andromeda.rutgers.edu/~feighn/papers/outer.pdf )

Benoist, Y.; Quint, J.-F. Stationary measures and invariant subsets of homogeneous spaces II, J. Amer. Math. Soc., Volume 26 (2013), pp. 659-734 | DOI | MR | Zbl

Castaing, C. Sur les multi-applications mesurables, Modélisation Mathématique et Analyse Numérique, Volume 1 (1967), pp. 91-126 | Numdam | MR | Zbl

Cohen, M.; Lustig, M. Very small group actions on $ℝ$ -trees and Dehn twist automorphisms, Topology, Volume 34 (1995), pp. 575-617 | DOI | MR | Zbl

Calegari, D.; Maher, J. Statistics and compression of scl, Erg. Theo. Dyn. Syst., Volume 35 (2015), pp. 64-110 | DOI | MR | Zbl

Culler, M.; Morgan, J. Group actions on $ℝ$ -trees, Proc. London Math. Soc., Volume 55 (1987), pp. 571-604 | DOI | MR | Zbl

Castaing, C.; Valadier, M., Lecture Notes in Math., 580, Springer, 1977 | MR | Zbl

Culler, M.; Vogtmann, K. Moduli of graphs and automorphisms of free groups, Invent. math., Volume 84 (1986), pp. 91-119 | DOI | MR | Zbl

Culler, M.; Vogtmann, K. The boundary of outer space in rank two, Arboreal group theory (Berkeley, CA, 1988) (Math. Sci. Res. Inst. Publ.), Volume 19, Springer, New York (1991), pp. 189-230 | DOI | MR | Zbl

Furstenberg, H.; Kesten, H. Products of Random Matrices, Ann. Math. Statist., Volume 31 (1960), pp. 457-469 | DOI | MR | Zbl

Furstenberg, H.; Kifer, Y. Random matrix products and measures on projective spaces, Israel J. Math., Volume 46 (1983), pp. 12-32 | DOI | MR | Zbl

Fathi, A.; Laudenbach, F.; Poenaru, V. Travaux de Thurston sur les surfaces, Astérisque, Volume 66-67 (1979) | Numdam | MR

Francaviglia, S.; Martino, A. Stretching factors, metrics and train tracks for free products (preprint arXiv:1312.4172 ) | MR

Francaviglia, S.; Martino, A. Metric Properties of Outer Space, Publ. Mat., Volume 55 (2011), pp. 433-473 | DOI | MR | Zbl

Furstenberg, H. Non commuting random products, Trans. Amer. Math. Soc., Volume 108 (1963), pp. 377-428 | DOI | MR | Zbl

Furstenberg, H., Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math.), Volume XXVI, 1973, pp. 193-229 | DOI | MR | Zbl

Guirardel, V.; Levitt, G. JSJ decompositions : definitions, existence, uniqueness. I: The JSJ deformation space (preprint arXiv:0911.3173 )

Guirardel, V.; Levitt, G. The outer space of a free product, Proc. London Math. Soc., Volume 94 (2007), pp. 695-714 | DOI | MR | Zbl

Gaboriau, D.; Levitt, G. The rank of actions on $ℝ$ -trees, Ann. Scient. Éc. Norm. Sup., Volume 28 (1995), pp. 549-570 | DOI | Numdam | MR | Zbl

Gromov, M. Hyperbolic manifolds, groups and actions, Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (Kra, I.; Maskit, B., eds.), Princeton Univ. Press (1980), pp. 183-213 | MR | Zbl

Guirardel, V. Dynamics of $Out (F_{n})$ on the boundary of outer space, Ann. Scient. Éc. Norm. Sup., Volume 33 (2000), pp. 433-465 | DOI | Numdam | MR | Zbl

Guirardel, V. Limit groups and groups acting freely on $ℝ^{n}$ -trees, Geom. Topol., Volume 8 (2004), pp. 1427-1470 | DOI | MR | Zbl

Guirardel, V. Actions of finitely generated groups on $ℝ$ -trees, Ann. Inst. Fourier, Volume 58 (2008), pp. 159-211 | DOI | Numdam | MR | Zbl

Guirardel, V. Approximations of stable actions on $ℝ$ -trees, Comment. Math. Helv., Volume 73 (1998), pp. 89-121 | DOI | MR | Zbl

Hatcher, A. Homological stability for automorphism groups of free groups, Comment. Math. Helv., Volume 70 (1995), pp. 39-62 | DOI | MR | Zbl

Hennion, H. Loi des grands nombres et perturbations pour des produits réductibles de matrices aléatoires indépendantes, Z. Wahrsch. verw. Gebiete, Volume 67 (1984), pp. 265-278 | DOI | MR | Zbl

Handel, M.; Mosher, L. Subgroup classification in $Out (F_{n})$ (preprint arXiv:0908.1255 )

Horbez, C. The boundary of the outer space of a free product (preprint arXiv:1408.0543 ) | MR

Horbez, C. Sphere paths in outer space, Alg. Geom. Top., Volume 12 (2012), pp. 2493-2517 | DOI | MR | Zbl

Horbez, C. A short proof of Handel and Mosher's alternative for subgroups of $Out (F_{N})$ , Groups Geom. Dyn., Volume 10 (2014), pp. 709-721 | DOI | MR

Horbez, C. Spectral rigidity for primitive elements of $F_{N}$ , J. Group Theory, Volume 19 (2016), pp. 55-123 | DOI | MR

Horbez, C. The Poisson boundary of $Out (F_{N})$ , Duke Math. J., Volume 165 (2016), pp. 341-369 | MR

Hatcher, A.; Vogtmann, K. The complex of free factors of a free group, Quart. J. Math. Oxford Ser., Volume 49 (1998), pp. 459-468 | DOI | MR | Zbl

Jiang, R. Branch Points and Free Actions On $ℝ$ -Trees, Arboreal Group Theory (Alperin, R., ed.) (Mathematical Sciences Research Institute Publications), Volume 19, Springer (1991), pp. 251-293 | DOI | MR | Zbl

Kaimanovich, V. Poisson boundaries of random walks on discrete solvable groups, Proceedings of Conference on Probability Measures on Groups X (Oberwolfach) (Heyer, H., ed.), Plenum, New York (1991), pp. 205-238 | DOI | MR | Zbl

Kapovich, I. The frequency space of a free group, Internat. J. Alg. Comput., Volume 15 (2005), pp. 939-969 | DOI | MR | Zbl

Kapovich, I. Currents on free groups, Topological and Asymptotic Aspects of Group Theory (Grigorchuk, R.; Mihalik, M.; Sapir, M.; Sunik, Z., eds.) (AMS Contemp. Math. Series), Volume 394 (2006), pp. 149-176 | DOI | MR | Zbl

Karlsson, A. Two extensions of Thurston's spectral theorem for surface diffeomorphisms, Bull. London Math. Soc., Volume 46 (2014), pp. 217-226 | DOI | MR | Zbl

Kingman, J. The Ergodic Theory of Subadditive Stochastic Processes, J. Roy. Statist. Soc. B, Volume 30 (1968), pp. 499-510 | DOI | MR | Zbl

Karlsson, A.; Ledrappier, F. On laws of large numbers for random walks, Ann. Probab., Volume 34 (2006), pp. 1693-1706 | DOI | MR | Zbl

Kapovich, I.; Lustig, M. The actions of $Out (F_{k})$ on the boundary of Outer space and on the space of currents: minimal sets and equivariant incompatibility, Erg. Th. Dyn. Syst., Volume 27 (2007), pp. 827-847 | DOI | MR | Zbl

Kapovich, I.; Lustig, M. Geometric intersection number and analogues of the curve complex for free groups, Geom. Topol., Volume 13 (2009), pp. 1805-1833 | DOI | MR | Zbl

Karlsson, A.; Ledrappier, F. Noncommutative Ergodic Theorems, Geometry, Rigidity and Group Actions (Farb, B.; Fisher, D., eds.) (Chicago Lectures in Math) (2011), pp. 396-418 | MR | Zbl

Kapovich, I.; Levitt, G.; Schupp, P.; Shpilrain, V. Translation equivalence in free groups, Trans. Amer. Math. Soc., Volume 359 (2007), pp. 1527-1546 | DOI | MR | Zbl

Kaimanovich, V.; Masur, H. The Poisson boundary of the mapping class group, Invent. math., Volume 125 (1996), pp. 221-264 | DOI | MR | Zbl

Levitt, G. Counting growth types of automorphisms of free groups, Geom. Funct. Anal., Volume 19 (2009), pp. 1119-1146 | DOI | MR | Zbl

Levitt, G.; Paulin, F. Geometric group actions on trees, Amer. J. Math., Volume 119 (1997), pp. 83-102 | DOI | MR | Zbl

Maher, J. Random walks on the mapping class group, Duke Math. J., Volume 156 (2011), pp. 429-468 | DOI | MR | Zbl

Martin, R. Non-Uniquely Ergodic Foliations of Thin Type, Measured Currents and Automorphisms of Free Groups (1995) | MR

Meinert, S. The Lipschitz metric on deformation spaces of $G$ -trees, Alg. Geom. Topol., Volume 15 (2015), pp. 987-1029 | DOI | MR | Zbl

Maher, J.; Tiozzo, G. Random walks on weakly hyperbolic groups (preprint arXiv:1410.4173 ) | MR

Reynolds, P. On indecomposable trees in the boundary of Outer space, Geom. Dedic., Volume 153 (2011), pp. 59-71 | DOI | MR | Zbl

Rieffel, M. Group $C^{*}$ -algebras as compact quantum metric spaces, Documenta Math., Volume 7 (2002), pp. 605-651 | DOI | MR | Zbl

Shenitzer, A. Decomposition of a group with a single defining relation into a free product, Proc. Amer. Math. Soc., Volume 6 (1955), pp. 273-279 | DOI | MR | Zbl

Stallings, J. Foldings of $G$ -trees, Arboreal Group Theory (Alperin, R., ed.) (Math. Sci. Res. Inst. Publ.), Volume 19, Springer, New York, Inc. (1991), pp. 355-368 | DOI | MR | Zbl

Swarup, G. Decompositions of free groups, J. Pure Appl. Algebra, Volume 40 (1986), pp. 99-102 | DOI | MR | Zbl

Vogtmann, K. Automorphisms of Free Groups and Outer Space, Geom. Dedic., Volume 94 (2002), pp. 1-31 | DOI | MR | Zbl

Walsh, C. The horoboundary and isometry group of Thurston's Lipschitz metric, Handbook of Teichmüller theory (Papadopoulos, A., ed.), Volume IV (2014), pp. 327-353 | DOI | MR | Zbl

Woess, W. Boundaries of random walks on graphs and groups with infinitely many ends, Israel J. Math., Volume 68 (1989), pp. 271-301 | DOI | MR | Zbl

Cité par Sources :