## Duke Mathematical Journal

### The Poisson boundary of $\operatorname{Out}(F_{N})$

Camille Horbez

#### Abstract

Let $\mu$ be a probability measure on $\operatorname {Out}(F_{N})$ with finite first logarithmic moment with respect to the word metric, finite entropy, and whose support generates a nonelementary subgroup of $\operatorname {Out}(F_{N})$. We show that almost every sample path of the random walk on $(\operatorname {Out}(F_{N}),\mu)$, 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 $(\operatorname {Out}(F_{N}),\mu)$. Using Bestvina and Reynolds’s 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 $\mathcal{FF}_{\mathcal{N}}$ 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 $(\operatorname {Out}(F_{N}),\mu)$.

#### Article information

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

Dates
Revised: 1 March 2015
First available in Project Euclid: 19 January 2016

https://projecteuclid.org/euclid.dmj/1453211877

Digital Object Identifier
doi:10.1215/00127094-3166308

Mathematical Reviews number (MathSciNet)
MR3457676

Zentralblatt MATH identifier
06556670

#### Citation

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

#### References

• [1] Y. Algom-Kfir, Strongly contracting geodesics in outer space, Geom. Topol. 15 (2011), 2181–2233.
• [2] Y. Algom-Kfir, The metric completion of outer space, preprint, arXiv:1202.6392v4 [math.GR].
• [3] W. Ballmann, On the Dirichlet problem at infinity for manifolds of nonpositive curvature, Forum Math. 1 (1989), 201–213.
• [4] M. Bestvina and M. Feighn, Hyperbolicity of the complex of free factors, Adv. Math. 256 (2014), 104–155.
• [5] M. Bestvina and M. Feighn, Outer limits, preprint, 1994.
• [6] M. Bestvina and P. Reynolds, The boundary of the complex of free factors, preprint, arXiv:1211.3608v2 [math.GR].
• [7] P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968.
• [8] G. D. Birkhoff, Proof of the ergodic theorem, Proc. Natl. Acad. Sci USA 17 (1931), 656–660.
• [9] F. Bonahon, “Geodesic currents on negatively curved groups” in Arboreal Group Theory (Berkeley, Calif., 1988), Math. Sci. Res. Inst. Publ. 19, Springer, New York, 1991, 143–168.
• [10] D. Calegari and J. Maher, Statistics and compression of scl, Ergodic Theory Dynam. Systems 35 (2015), 64–110.
• [11] CH. Castaing, Sur les multi-applications mesurables, Rev. Française Informat. Recherche Opérationnell 1 (1967), 91–126.
• [12] M. M. Cohen and M. Lustig, Very small group actions on $\mathbb{R}$-trees and Dehn twist automorphisms, Topology 34 (1995), 575–617.
• [13] Y. Coudène, Théorie ergodique et systèmes dynamiques, EDP Sciences, Les Ulis, CNRS Éditions, Paris, 2012.
• [14] T. Coulbois and A. Hilion, Ergodic currents dual to a real tree, preprint, arXiv:1302.3766v2 [math.GR].
• [15] T. Coulbois, A. Hilion, and M. Lustig, Non-unique ergodicity, observers’ topology and the dual algebraic lamination for $\mathbb{R}$-trees, Illinois J. Math. 51 (2007), 897–911.
• [16] T. Coulbois, A. Hilion, and M. Lustig, $\mathbb{R}$-trees and laminations for free groups, I: Algebraic laminations, J. Lond. Math. Soc. (2) 78 (2008), 723–736.
• [17] T. Coulbois, A. Hilion, and M. Lustig, $\mathbb{R}$-trees and laminations for free groups, II: The dual lamination of an $\mathbb{R}$-tree, J. Lond. Math. Soc. (2) 78 (2008), 737–754.
• [18] M. Culler and J. W. Morgan, Group actions on $\mathbb{R}$-trees, Proc. Lond. Math. Soc. (3) 55 (1987), 571–604.
• [19] M. Culler and K. Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), 91–119.
• [20] S. Francaviglia and A. Martino, Metric properties of outer space, Publ. Mat. 55 (2011), 433–473.
• [21] H. Furstenberg, “Boundary theory and stochastic processes on homogeneous spaces” in Harmonic Analysis on Homogeneous Spaces (Williamstown, Mass., 1972), Proc. Sympos. Pure Math. XXVI, Amer. Math. Soc., Providence, 1973, 193–229.
• [22] V. Guirardel, Dynamics of $\operatorname{Out}({F}_{n})$ on the boundary of outer space, Ann. Sci. Éc. Norm. Supér. (4) 33 (2000), 433–465.
• [23] U. Hamenstädt, Lines of minima in outer space, Duke Math. J. 163 (2014), 733–776.
• [24] U. Hamenstädt, The boundary of the free splitting graph and the free factor graph, preprint, arXiv:1211.1630v5 [math.GT].
• [25] M. Handel and L. Mosher, Subgroup classification in $\operatorname{Out} ({F}_{n})$, preprint, arXiv:0908.1255v1 [math.GR].
• [26] A. Hatcher and K. Vogtmann, The complex of free factors of a free group, Quart. J. Math. Oxford Ser. (2) 49 (1998), 459–468.
• [27] C. Horbez, A short proof of Handel and Mosher’s alternative for subgroups of $\operatorname{Out}({F}_{N})$, preprint, arXiv:1404.4626v1 [math.GR].
• [28] V. A. Kaimanovich, “Measure-theoretic boundaries of Markov chains, 0–2 laws and entropy” in Harmonic Analysis and Discrete Potential Theory (Frascati, 1991), Plenum, New York, 1992, 145–180.
• [29] V. A. Kaimanovich, The Poisson formula for groups with hyperbolic properties, Ann. of Math. (2) 152 (2000), 659–692.
• [30] V. A. Kaimanovich and H. Masur, The Poisson boundary of the mapping class group, Invent. Math. 125 (1996), 221–264.
• [31] I. Kapovich, The frequency space of a free group, Internat. J. Algebra Comput. 15 (2005), 939–969.
• [32] I. Kapovich, “Currents on free groups” in Topological and Asymptotic Aspects of Group Theory, Contemp. Math. 394, Amer. Math. Soc., Providence, 2006, 149–176.
• [33] I. Kapovich and M. Lustig, Geometric intersection number and analogues of the curve complex for free groups, Geom. Topol. 13 (2009), 1805–1833.
• [34] I. Kapovich and M. Lustig, Intersection form, laminations and currents on free groups, Geom. Funct. Anal. 19 (2010), 1426–1467.
• [35] I. Kapovich and M. Lustig, Stabilizers of $\mathbb{R}$-trees with free isometric actions of ${F}_{n}$, J. Group Theory 14 (2011), 673–694.
• [36] I. Kapovich and K. Rafi, On hyperbolicity of free splitting and free factor complexes, Groups Geom. Dyn. 8 (2014), 391–414.
• [37] J. Maher and G. Tiozzo, Random walks on weakly hyperbolic groups, preprint, arXiv:1410.4173v2 [math.GT].
• [38] R. Martin, Non-uniquely ergodic foliations of thin-type, measured currents and automorphisms of free groups, Ph.D. dissertation, University of California, Los Angeles, Calif., 1995.
• [39] H. Namazi, A. Pettet, and P. Reynolds, Ergodic decompositions for folding and unfolding paths in outer space, preprint, arXiv:1410.8870v1 [math.GT].
• [40] R. R. Phelps, ed., Lectures on Choquet’s Theorem, 2nd ed., Lecture Notes in Math. 1757, Springer, Berlin, 1966.
• [41] P. Reynolds, On indecomposable trees in the boundary of outer space, Geom. Dedicata 153 (2011), 59–71.
• [42] P. Reynolds, Reducing systems for very small trees, preprint, arXiv:1211.3378v1 [math.GR].
• [43] V. A. Rokhlin, On the fundamental ideas of measure theory, Mat. Sb. (N.S.) 25 (1949), no. 67, 107–150.
• [44] C. Uyanik, Generalized north–south dynamics on the space of geodesic currents, preprint, arXiv:1311.1470v3 [math.GT].
• [45] W. Woess, Boundaries of random walks on graphs and groups with infinitely many ends, Israel J. Math. 68 (1989), 271–301.