The horoboundary of outer space, and growth under random automorphisms

Horbez, Camille

doi:10.24033/asens.2304

Geodesic

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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.

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Abstract (VO)
Résumé

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.

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.

Publié le : 2016-05-27

DOI : 10.24033/asens.2304

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

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     title = {The horoboundary of outer space, and growth under random automorphisms},
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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://geodesic.mathdoc.fr/articles/10.24033/asens.2304/

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