Central limit theorems for mapping class groups and Out(FN)

Horbez, Camille¹

¹ Laboratoire de Mathématiques d’Orsay, CNRS - Université Paris Sud, Orsay, France

Geometry & topology, Tome 22 (2018) no. 1, pp. 105-156.

Résumé

We prove central limit theorems for the random walks on either the mapping class group of a closed, connected, orientable, hyperbolic surface, or on $Out\left(F_N\right)$, each time under a finite second moment condition on the measure (either with respect to the Teichmüller metric, or with respect to the Lipschitz metric on outer space). In the mapping class group case, this describes the spread of the hyperbolic length of a simple closed curve on the surface after applying a random product of mapping classes. In the case of $Out\left(F_N\right)$, this describes the spread of the length of primitive conjugacy classes in $F_N$ under random products of outer automorphisms. Both results are based on a general criterion for establishing a central limit theorem for the Busemann cocycle on the horoboundary of a metric space, applied to either the Teichmüller space of the surface or to the Culler–Vogtmann outer space.

Détail

DOI : 10.2140/gt.2018.22.105

Classification : 20F65, 60B15
Keywords: mapping class groups, Out(Fn), outer automorphism groups, random walks on groups, central limit theorem

Bibliographie

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

[2] Y Algom-Kfir, M Bestvina, Asymmetry of outer space, Geom. Dedicata 156 (2012) 81 | DOI

[3] Y Benoist, J F Quint, Central limit theorem for linear groups, Ann. Probab. 44 (2016) 1308 | DOI

[4] Y Benoist, J F Quint, Central limit theorem on hyperbolic groups, Izv. Ross. Akad. Nauk Ser. Mat. 80 (2016) 5 | DOI

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

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

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

[8] M Björklund, Central limit theorems for Gromov hyperbolic groups, J. Theoret. Probab. 23 (2010) 871 | DOI

[9] B M Brown, Martingale central limit theorems, Ann. Math. Statist. 42 (1971) 59 | DOI

[10] Y E Choi, K Rafi, Comparison between Teichmüller and Lipschitz metrics, J. Lond. Math. Soc. 76 (2007) 739 | DOI

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

[12] M Culler, J W Morgan, Group actions on R–trees, Proc. London Math. Soc. 55 (1987) 571 | DOI

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

[14] F Dahmani, C Horbez, Spectral theorems for random walks on mapping class groups and Out(FN), preprint (2015)

[15] S Dowdall, M Duchin, H Masur, Statistical hyperbolicity in Teichmüller space, Geom. Funct. Anal. 24 (2014) 748 | DOI

[16] S Dowdall, S J Taylor, Hyperbolic extensions of free groups, preprint (2014)

[17] A Fathi, F Laudenbach, V Poenaru, editors, Travaux de Thurston sur les surfaces, 66, Soc. Math. France (1979) 284

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

[19] H Furstenberg, Noncommuting random products, Trans. Amer. Math. Soc. 108 (1963) 377 | DOI

[20] H Furstenberg, H Kesten, Products of random matrices, Ann. Math. Statist. 31 (1960) 457 | DOI

[21] F P Gardiner, H Masur, Extremal length geometry of Teichmüller space, Complex Variables Theory Appl. 16 (1991) 209 | DOI

[22] É Ghys, P De La Harpe, editors, Sur les groupes hyperboliques d’après Mikhael Gromov, 83, Birkhäuser (1990) | DOI

[23] I Y Gol’Dsheĭd, G A Margulis, Lyapunov exponents of a product of random matrices, Uspekhi Mat. Nauk 44 (1989) 13

[24] V Guirardel, Dynamics of Out(Fn) on the boundary of outer space, Ann. Sci. École Norm. Sup. 33 (2000) 433 | DOI

[25] Y Guivarc’H, A Raugi, Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence, Z. Wahrsch. Verw. Gebiete 69 (1985) 187 | DOI

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

[27] C Horbez, The boundary of the outer space of a free product, preprint (2014)

[28] C Horbez, The horoboundary of outer space, and growth under random automorphisms, Ann. Sci. Éc. Norm. Supér. 49 (2016) 1075 | DOI

[29] C Horbez, Spectral rigidity for primitive elements of FN, J. Group Theory 19 (2016) 55 | DOI

[30] V A Kaimanovich, H Masur, The Poisson boundary of the mapping class group, Invent. Math. 125 (1996) 221 | DOI

[31] I Kapovich, Currents on free groups, from: "Topological and asymptotic aspects of group theory" (editors R Grigorchuk, M Mihalik, M Sapir, Z Šunik), Contemp. Math. 394, Amer. Math. Soc. (2006) 149 | DOI

[32] I Kapovich, M Lustig, Geometric intersection number and analogues of the curve complex for free groups, Geom. Topol. 13 (2009) 1805 | DOI

[33] A Karlsson, Two extensions of Thurston’s spectral theorem for surface diffeomorphisms, Bull. Lond. Math. Soc. 46 (2014) 217 | DOI

[34] S P Kerckhoff, The asymptotic geometry of Teichmüller space, Topology 19 (1980) 23 | DOI

[35] J F C Kingman, The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. Ser. B 30 (1968) 499

[36] E Klarreich, The boundary at infinity of the curve complex and the relative Teichmüller space, preprint (1999)

[37] É Le Page, Théorèmes limites pour les produits de matrices aléatoires, from: "Probability measures on groups" (editor H Heyer), Lecture Notes in Math. 928, Springer (1982) 258

[38] A Lenzhen, K Rafi, J Tao, Bounded combinatorics and the Lipschitz metric on Teichmüller space, Geom. Dedicata 159 (2012) 353 | DOI

[39] L Liu, W Su, The horofunction compactification of the Teichmüller metric, from: "Handbook of Teichmüller theory, IV" (editor A Papadopoulos), IRMA Lect. Math. Theor. Phys. 19, Eur. Math. Soc. (2014) 355 | DOI

[40] J Maher, G Tiozzo, Random walks on weakly hyperbolic groups, preprint (2015)

[41] R Martin, Non-uniquely ergodic foliations of thin type, Ergodic Theory Dynam. Systems 17 (1997) 667 | DOI

[42] H A Masur, Y N Minsky, Geometry of the complex of curves, I : Hyperbolicity, Invent. Math. 138 (1999) 103 | DOI

[43] P Mathieu, A Sisto, Deviation inequalities for random walks, preprint (2014)

[44] Y N Minsky, Extremal length estimates and product regions in Teichmüller space, Duke Math. J. 83 (1996) 249 | DOI

[45] H Miyachi, Teichmüller rays and the Gardiner–Masur boundary of Teichmüller space, II, Geom. Dedicata 162 (2013) 283 | DOI

[46] H Miyachi, Unification of extremal length geometry on Teichmüller space via intersection number, Math. Z. 278 (2014) 1065 | DOI

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

[48] K Rafi, Thick-thin decomposition for quadratic differentials, Math. Res. Lett. 14 (2007) 333 | DOI

[49] S Sawyer, T Steger, The rate of escape for anisotropic random walks in a tree, Probab. Theory Related Fields 76 (1987) 207 | DOI

[50] C Walsh, The horoboundary and isometry group of Thurston’s Lipschitz metric, from: "Handbook of Teichmüller theory, IV" (editor A Papadopoulos), IRMA Lect. Math. Theor. Phys. 19, Eur. Math. Soc. (2014) 327 | DOI