Geodesic
Random trees in the boundary of outer space
Kapovich, Ilya1 ; Maher, Joseph2 ; Pfaff, Catherine3 ; Taylor, Samuel J4
1 Department of Mathematics and Statistics, Hunter College of CUNY, New York, NY, United States
2 Department of Mathematics, CUNY College of Staten Island and CUNY Graduate Center, Staten Island, NY, United States
3 Department of Mathematics and Statistics, Queen’s University, Kingston, ON, Canada
4 Department of Mathematics, Temple University, Philadelphia, PA, United States
Geometry & topology, Tome 26 (2022) no. 1, pp. 127-162.

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

We prove that for the harmonic measure associated to a random walk on Out ⁡ (Fr) satisfying some mild conditions, a typical tree in the boundary of Outer space is trivalent and nongeometric. This result answers a question of Mladen Bestvina.

Keywords: free group, random walk, outer space, free group automorphisms, train track maps

Kapovich, Ilya 1 ; Maher, Joseph 2 ; Pfaff, Catherine 3 ; Taylor, Samuel J 4

1 Department of Mathematics and Statistics, Hunter College of CUNY, New York, NY, United States
2 Department of Mathematics, CUNY College of Staten Island and CUNY Graduate Center, Staten Island, NY, United States
3 Department of Mathematics and Statistics, Queen’s University, Kingston, ON, Canada
4 Department of Mathematics, Temple University, Philadelphia, PA, United States 
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Kapovich, Ilya; Maher, Joseph; Pfaff, Catherine; Taylor, Samuel J. Random trees in the boundary of outer space. Geometry & topology, Tome 26 (2022) no. 1, pp. 127-162. http://geodesic.mathdoc.fr/item/GT_2022_26_1_a3/

[1] Y Algom-Kfir, Strongly contracting geodesics in outer space, Geom. Topol. 15 (2011) 2181 | DOI

[2] Y Algom-Kfir, I Kapovich, C Pfaff, Stable strata of geodesics in outer space, Int. Math. Res. Not. 2019 (2019) 4549 | DOI

[3] M. Bestvina, M. Feighn, Outer limits, preprint (1994)

[4] M Bestvina, M Feighn, Hyperbolicity of the complex of free factors, Adv. Math. 256 (2014) 104 | DOI

[5] M Bestvina, M Feighn, M Handel, Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal. 7 (1997) 215 | DOI

[6] M Bestvina, C Horbez, R D Wade, On the topological dimension of the Gromov boundaries of some hyperbolic Out(FN)–graphs, Pacific J. Math. 308 (2020) 1 | DOI

[7] M Bestvina, P Reynolds, The boundary of the complex of free factors, Duke Math. J. 164 (2015) 2213 | DOI

[8] M M Cohen, M Lustig, Very small group actions on R–trees and Dehn twist automorphisms, Topology 34 (1995) 575 | DOI

[9] T Coulbois, A Hilion, Botany of irreducible automorphisms of free groups, Pacific J. Math. 256 (2012) 291 | DOI

[10] T Coulbois, A Hilion, M Lustig, R–trees and laminations for free groups, I : Algebraic laminations, J. Lond. Math. Soc. 78 (2008) 723 | DOI

[11] T Coulbois, A Hilion, M Lustig, R–trees and laminations for free groups, II : The dual lamination of an R–tree, J. Lond. Math. Soc. 78 (2008) 737 | DOI

[12] M Culler, K Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986) 91 | DOI

[13] F Dahmani, C Horbez, Spectral theorems for random walks on mapping class groups and out (FN), Int. Math. Res. Not. 2018 (2018) 2693 | DOI

[14] S Dowdall, S J Taylor, The co-surface graph and the geometry of hyperbolic free group extensions, J. Topol. 10 (2017) 447 | DOI

[15] S Dowdall, S J Taylor, Hyperbolic extensions of free groups, Geom. Topol. 22 (2018) 517 | DOI

[16] S Francaviglia, A Martino, Metric properties of outer space, Publ. Mat. 55 (2011) 433 | DOI

[17] D Gaboriau, G Levitt, The rank of actions on R–trees, Ann. Sci. École Norm. Sup. 28 (1995) 549 | DOI

[18] V Gadre, J Maher, Recurrence of quadratic differentials for harmonic measure, Math. Proc. Cambridge Philos. Soc. 169 (2020) 299 | DOI

[19] U Hamenstaedt, The boundary of the free splitting graph and the free factor graph, preprint (2012)

[20] M Handel, L Mosher, Axes in outer space, 1004, Amer. Math. Soc. (2011) | DOI

[21] C Horbez, The Poisson boundary of Out(FN), Duke Math. J. 165 (2016) 341 | DOI

[22] C Horbez, Central limit theorems for mapping class groups and Out(FN), Geom. Topol. 22 (2018) 105 | DOI

[23] I Kapovich, J Maher, C Pfaff, S J Taylor, Random outer automorphisms of free groups: attracting trees and their singularity structures, Trans. Amer. Math. Soc. 375 (2022) 525 | DOI

[24] J Maher, G Tiozzo, Random walks on weakly hyperbolic groups, J. Reine Angew. Math. 742 (2018) 187 | DOI

[25] L Mosher, C Pfaff, Lone axes in outer space, Algebr. Geom. Topol. 16 (2016) 3385 | DOI

[26] H Namazi, A Pettet, P Reynolds, Ergodic decompositions for folding and unfolding paths in outer space, preprint (2014)

[27] F Paulin, The Gromov topology on R–trees, Topology Appl. 32 (1989) 197 | DOI

[28] P Reynolds, Reducing systems for very small trees, preprint (2012)

[29] S J Taylor, G Tiozzo, Random extensions of free groups and surface groups are hyperbolic, Int. Math. Res. Not. 2016 (2016) 294 | DOI