×

zbMATH — the first resource for mathematics

Random walks on weakly hyperbolic groups. (English) Zbl 1434.60015
Summary: Let \(G\) be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space \(X\). We say the action of \(G\) is weakly hyperbolic if \(G\) contains two independent hyperbolic isometries. We show that a random walk on such \(G\) converges to the Gromov boundary almost surely. We apply the convergence result to show linear progress and linear growth of translation length, without any assumptions on the moments of the random walk.
If the action is acylindrical, and the random walk has finite entropy and finite logarithmic moment, we show that the Gromov boundary with the hitting measure is the Poisson boundary.

MSC:
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
20F65 Geometric group theory
60G50 Sums of independent random variables; random walks
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] C. D. Aliprantis and K. C. Border, Infinite dimensional analysis, 3rd ed., Springer, Berlin 2006. · Zbl 1156.46001
[2] U. Bader, B. Duchesne and J. Lécureux, Furstenberg maps for CAT(0) targets of finite telescopic dimension, preprint (2014), ; to appear in Ergodic Theory Dynam. Systems. · Zbl 1378.37055
[3] W. Ballmann, M. Gromov and V. Schroeder, Manifolds of nonpositive curvature, Progr. Math. 61, Birkhäuser, Boston 1985. · Zbl 0591.53001
[4] M. Bestvina, K. Bromberg and K. Fujiwara, Constructing group actions on quasi-trees and applications to mapping class groups, Publ. Math. Inst. Hautes Études Sci. 122 (2015), 1-64. · Zbl 1372.20029
[5] M. Bestvina and M. Feighn, A hyperbolic {{\rm Out}(F_{n})}-complex, Groups Geom. Dyn. 4 (2010), no. 1, 31-58. · Zbl 1190.20017
[6] M. Bestvina and M. Feighn, Hyperbolicity of the complex of free factors, Adv. Math. 256 (2014), 104-155. · Zbl 1348.20028
[7] M. Björklund, Central limit theorems for gromov hyperbolic groups, J. Theoret. Probab. 23 (2010), no. 3, 871-887. · Zbl 1217.60019
[8] S. Blachère, P. Haïssinsky and P. Mathieu, Harmonic measures versus quasiconformal measures for hyperbolic groups, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 4, 683-721. · Zbl 1243.60005
[9] M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000), no. 2, 266-306. · Zbl 0972.53021
[10] B. H. Bowditch, Tight geodesics in the curve complex, Invent. Math. 171 (2008), no. 2, 281-300. · Zbl 1185.57011
[11] M. R. Bridson, On the dimension of CAT(0) spaces where mapping class groups act, J. reine angew. Math. 673 (2012), 55-68. · Zbl 1270.20046
[12] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren Math. Wiss. 319, Springer, Berlin 1999.
[13] D. Calegari and J. Maher, Statistics and compression of scl, Ergodic Theory Dynam. Systems 35 (2015), no. 1, 64-110. · Zbl 1351.37214
[14] T. Delzant and P. Py, Kähler groups, real hyperbolic spaces and the Cremona group, Compos. Math. 148 (2012), no. 1, 153-184. · Zbl 1326.32038
[15] M. Duchin, Geodesics track random walks in Teichmüller space, ProQuest LLC, Ann Arbor 2005; Ph.D. thesis, The University of Chicago, 2005.
[16] K. Fujiwara, Subgroups generated by two pseudo-Anosov elements in a mapping class group. I: Uniform exponential growth, Groups of diffeomorphisms, Adv. Stud. Pure Math. 52, Mathematical Society of Japan, Tokyo (2008), 283-296. · Zbl 1170.57017
[17] H. Furstenberg, Noncommuting random products, Trans. Amer. Math. Soc. 108 (1963), 377-428. · Zbl 0203.19102
[18] F. Gautero and F. Mathéus, Poisson boundary of groups acting on {\mathbb{R}}-trees, Israel J. Math. 191 (2012), no. 2, 585-646. · Zbl 1293.20027
[19] Y. Guivarc’h, Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire, Conference on random walks (Kleebach 1979), Astérisque 74, Société Mathématique de France, Paris (1980), 47-98, 3. · Zbl 0448.60007
[20] U. Hamenstädt, Train tracks and the gromov boundary of the complex of curves, Spaces of Kleinian groups, London Math. Soc. Lecture Note Ser. 329, Cambridge University Press, Cambridge (2006), 187-207. · Zbl 1117.30036
[21] M. Handel and L. Mosher, The free splitting complex of a free group. I: Hyperbolicity, Geom. Topol. 17 (2013), no. 3, 1581-1672. · Zbl 1278.20053
[22] J. L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986), no. 1, 157-176. · Zbl 0592.57009
[23] C. Horbez, The Poisson boundary of {Out(F_{N})}, preprint (2014), ; to appear in Duke Math. J.
[24] V. A. Kaimanovich, Lyapunov exponents, symmetric spaces and a multiplicative ergodic theorem for semisimple Lie groups (Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 164 (1987), 29-46, 196-197; translation in J. Soviet Math. 47 (1989), no. 2, 2387-2398.
[25] V. A. Kaimanovich, The Poisson boundary of hyperbolic groups, C. R., Acad. Sci. Paris Sér. I Math. 318 (1994), no. 1, 59-64. · Zbl 0792.60006
[26] V. A. Kaimanovich, The Poisson formula for groups with hyperbolic properties, Ann. of Math. (2) 152 (2000), no. 3, 659-692. · Zbl 0984.60088
[27] V. A. Kaimanovich and H. A. Masur, The Poisson boundary of the mapping class group, Invent. Math. 125 (1996), no. 2, 221-264. · Zbl 0864.57014
[28] I. Kapovich and N. Benakli, Boundaries of hyperbolic groups, Combinatorial and geometric group theory (New York 2000, Hoboken 2001), Contemp. Math. 296, American Mathematical Society, Providence (2002), 39-93. · Zbl 1044.20028
[29] M. Kapovich and B. Leeb, Actions of discrete groups on nonpositively curved spaces, Math. Ann. 306 (1996), no. 2, 341-352. · Zbl 0856.20024
[30] A. Karlsson and F. Ledrappier, On laws of large numbers for random walks, Ann. Probab. 34 (2006), no. 5, 1693-1706. · Zbl 1111.60005
[31] A. Karlsson and F. Ledrappier, Linear drift and Poisson boundary for random walks, Pure Appl. Math. Q. 3 (2007), no. 4, 1027-1036. · Zbl 1142.60035
[32] S.-H. Kim and T. Koberda, Embedability between right-angled artin groups, Geom. Topol. 17 (2013), no. 1, 493-530. · Zbl 1278.20049
[33] S.-H. Kim and T. Koberda, The geometry of the curve graph of a right-angled Artin group, Internat. J. Algebra Comput. 24 (2014), no. 2, 121-169. · Zbl 1342.20042
[34] J. F. C. Kingman, The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. Ser. B 30 (1968), 499-510. · Zbl 0182.22802
[35] E. Klarreich, The boundary at infinity of the curve complex and the relative Teichmüller space, preprint (1998).
[36] E. Kowalski, The large sieve and its applications, Cambridge Tracts in Math. 175, Cambridge University Press, Cambridge 2008.
[37] F. Ledrappier, Some asymptotic properties of random walks on free groups, Topics in probability and Lie groups: Boundary theory, CRM Proc. Lecture Notes 28, American Mathematical Society, Providence (2001), 117-152. · Zbl 0994.60073
[38] J. Maher, Linear progress in the complex of curves, Trans. Amer. Math. Soc. 362 (2010), no. 6, 2963-2991. · Zbl 1232.37023
[39] J. Maher, Random Heegaard splittings, J. Topol. 3 (2010), no. 4, 997-1025. · Zbl 1207.37027
[40] J. Maher, Random walks on the mapping class group, Duke Math. J. 156 (2011), no. 3, 429-468. · Zbl 1213.37072
[41] J. Maher, Exponential decay in the mapping class group, J. Lond. Math. Soc. (2) 86 (2012), no. 2, 366-386. · Zbl 1350.37010
[42] A. Malyutin and P. Svetlov, Poisson-Furstenberg boundaries of fundamental groups of closed 3-manifolds, preprint (2014), .
[43] Y. I. Manin, Cubic forms: Algebra, geometry, arithmetic, North-Holland Math. Library 4, North-Holland Publishing, Amsterdam 1974.
[44] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergeb. Math. Grenzgeb. (3) 17, Springer, Berlin 1991. · Zbl 0732.22008
[45] H. A. Masur and Y. N. Minsky, Geometry of the complex of curves. I: Hyperbolicity, Invent. Math. 138 (1999), no. 1, 103-149. · Zbl 0941.32012
[46] P. Mathieu and A. Sisto, Deviation inequalities for random walks, preprint (2014), .
[47] A. Nevo and M. Sageev, The Poisson boundary of {{\rm CAT}(0)} cube complex groups, Groups Geom. Dyn. 7 (2013), no. 3, 653-695. · Zbl 1346.20084
[48] V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179-210.
[49] D. Osin, Acylindrically hyperbolic groups, Trans. Amer. Math. Soc. 368 (2016), 851-888. · Zbl 1380.20048
[50] I. Rivin, Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms, Duke Math. J. 142 (2008), no. 2, 353-379. · Zbl 1207.20068
[51] Z. Sela, Acylindrical accessibility for groups, Invent. Math. 129 (1997), no. 3, 527-565. · Zbl 0887.20017
[52] A. Sisto, Contracting elements and random walks, preprint (2011), ; J. reine angew. Math. (2016), DOI 10.1515/crelle-2015-0093.
[53] A. Sisto, Tracking rates for random walks, preprint (2013), ; to appear in Israel J. Math.
[54] G. Tiozzo, Sublinear deviation between geodesics and sample paths, Duke Math. J. 164 (2015), no. 3, 511-539. · Zbl 1314.30085
[55] J. Väisälä, Gromov hyperbolic spaces, Expo. Math. 23 (2005), no. 3, 187-231. · Zbl 1087.53039
[56] W. Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Math. 138, Cambridge University Press, Cambridge 2000.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.