Abstract
We prove that almost every path of a random walk on a finitely generated nonamenable group converges in the compactification of the group introduced by W. J. Floyd. In fact, we consider the more general setting of ergodic cocycles of some semigroup of one-Lipschitz maps of a complete metric space with a boundary constructed following Gromov. We obtain in addition that when the Floyd boundary of a finitely generated group is non-trivial, then it is in fact maximal in the sense that it can be identified with the Poisson boundary of the group with reasonable measures. The proof relies on works of Kaimanovich together with visibility properties of Floyd boundaries. Furthermore, we discuss mean proximality of ϖΓ and a conjecture of McMullen. Lastly, related statements about the convergence of certain sequences of points, for example quasigeodesic rays or orbits of one-Lipschitz maps, are obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[A]Ancona, A., Théorie du potentiel sur les graphes et les variétés, inÉcole d’été de Probabilités de Saint-Flour XVIII, 1988 (Hennequin, P. L., ed.), Lecture Notes in Math.1427, pp. 1–112. Springer-Verlag, Berlin-Heidelberg, 1990.
[BHK]Bonk, M., Heinonen, J. andKoskela, P., Uniformizing Gromov hyperbolic space,Astérisque 270 (2001), 1–99.
[CS]Cartwright, D. I. andSoardi, P. M., Convergence to ends for random walks on the automorphism group of a tree,Proc. Amer. Math. Soc. 107 (1989), 817–823.
[CDP]Coornaert, M., Delzant, T. andPapadopoulos, A.,Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov. Lecture Notes in Math.1441, Springer-Verlag, Berlin-Heidelberg, 1990.
[E]Epstein, D. B. A., Cannon, J. W., Holt, D. F., Levy, S. V. F., Paterson, M. S. andThurston, W. P.,Word Processing in Groups, Jones and Bartlett, Boston, Mass., 1992.
[F1]Floyd, W. J., Group completions and limit sets of Kleinian groups,Invent. Math. 57 (1980), 205–218.
[F2]Floyd, W. J., Group completions and Furstenberg boundaries: rank one,Duke Math. J. 51 (1984), 1017–1020.
[Ful]Furstenberg, H., Random walks and discrete subgroups of Lie groups, inAdvances in Probability and Related Topics, vol.1 (Ney, P., ed.), pp. 1–63, Marcel Dekker, New York, 1971.
[Fu2]Furstenberg, H., Boundary theory and stochastic processes on homogeneous spaces, inHarmonic Analysis on Homogeneous Spaces (Williamstown, Mass., 1972) (Moore, C. C., ed.), Proc. Symp. Pure Math.26, pp. 193–229, Amer. Math. Soc., Providence, R. I., 1973.
[G]Gromov, M., Hyperbolic groups, inEssays in Group Theory (Gersten, S., ed.), Math. Sci. Res. Inst. Publ.8, pp. 75–263, Springer-Verlag, New York, 1987.
[Gu]Guivarc’h, Y., Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire,Astérisque 74 (1980), 47–98.
[Ka1]Kaimanovich, V. A., The Poisson boundary of hyperbolic groups,C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), 59–64.
[Ka2]Kaimanovich, V. A., The Poisson formula for groups with hyperbolic properties,Ann. of Math. 152 (2000), 659–692.
[K1]Karlsson, A., Non-expanding maps and Busemann functions,Ergodic Theory Dynam. Systems 21 (2001), 1447–1457.
[K2]Karlsson, A., Free subgroups of groups with non-trivial Floyd boundary, to appear inComm. Algebra.
[KM]Karlsson, A. andMargulis, G. A., A multiplicative ergodic theorem and nonpositively curved spaces,Comm. Math. Phys. 208 (1999), 107–123.
[M]Margulis, G. A.,Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, New York, 1991.
[Mc]McMullen, C. T., Local connectivity, Kleinian groups and geodesics on the blowup of the torus,Invent. Math. 146 (2001), 35–91.
[S]Stallings, J.,Group Theory and Three-Dimensional Manifolds, Yale Univ. Press, New Haven-London, 1971.
[St]Stark, C. W., Group completions and orbifolds of variable negative curvature,Proc. Amer. Math. Soc. 114 (1992), 191–194.
[T1]Tukia, P., A remark on a paper by Floyd, inHolomorphic Functions and Moduli, Vol. II (Berkeley, Calif., 1986) (Drasin, D., Earle, C. J., Gehring, F. W., Kra, I. and Marden, A., eds.), Math. Sci. Res. Inst. Publ.11, pp. 165–172, Springer-Verlag, New York, 1988.
[T2]Tukia, P., The limit map of a homomorphism of discrete Möbius groups,Inst. Hautes Études Sci. Publ. Math. 82 (1995), 97–132.
[W]Woess, W., Boundaries of random walks on graphs and groups with infinitely many ends,Israel J. Math. 68 (1989), 271–301.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Karlsson, A. Boundaries and random walks on finitely generated infinite groups. Ark. Mat. 41, 295–306 (2003). https://doi.org/10.1007/BF02390817
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02390817