Duke Mathematical Journal

The Poisson boundary of Out(FN)

Camille Horbez

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

Article information

Duke Math. J., Volume 165, Number 2 (2016), 341-369.

Received: 27 June 2014
Revised: 1 March 2015
First available in Project Euclid: 19 January 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization

$\operatorname {Out}(F_{N})$ random walks Poisson boundary


Horbez, Camille. The Poisson boundary of $\operatorname{Out}(F_{N})$. Duke Math. J. 165 (2016), no. 2, 341--369. doi:10.1215/00127094-3166308. https://projecteuclid.org/euclid.dmj/1453211877

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