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Article Dans Une Revue Duke Mathematical Journal Année : 2016

The Poisson boundary of $\text{Out}(F_N)$

Résumé

Let $\mu$ be a probability measure on $\text{Out}(F_N)$ with finite first logarithmic moment with respect to the word metric, finite entropy, and whose support generates a nonelementary subgroup of $\text{Out}(F_N)$. We show that almost every sample path of the random walk on $(\text{Out}(F_N),\mu)$, when realized in Culler and Vogtmann's outer space, converges to the simplex of a free, arational tree. We then prove that the space $\mathcal{FI}$ of simplices of free and arational trees, equipped with the hitting measure, is the Poisson boundary of $(\text{Out}(F_N),\mu)$. Using Bestvina-Reynolds' and Hamenstädt's description of the Gromov boundary of the complex $\mathcal{FF}_N$ of free factors of $F_N$, this gives a new proof of the fact, due to Calegari and Maher, that the realization in $\mathcal{FF_N}$ of almost every sample path of the random walk converges to a boundary point. We get in addition that $\partial\mathcal{FF}_N$, equipped with the hitting measure, is the Poisson boundary of $(\text{Out}(F_N),\mu)$.

Dates et versions

hal-00998397 , version 1 (02-06-2014)

Identifiants

Citer

Camille Horbez. The Poisson boundary of $\text{Out}(F_N)$. Duke Mathematical Journal, 2016, 165 (2), pp.341-369. ⟨10.1215/00127094-3166308⟩. ⟨hal-00998397⟩
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