Abstract
We introduce a method to estimate the entropy of random walks on groups. We apply this method to exhibit the existence of compact manifolds with amenable fundamental groups such that the universal cover is not Liouville. We also use the criterion to prove that a finitely generated solvable group admits a symmetric measure with non-trivial Poisson boundary if and only if this group is not virtually nilpotent. This, in particular, shows that any polycyclic group admits a symmetric measure such that its boundary does not readily interprete in terms of the ambient Lie group. As another application we get a series of examples of amenable groups such that any finite entropy non-degenerate measure on them has non-trivial Poisson boundary. Since the groups in question are amenable, they do admit measures such that the corresponding random walks have trivial boundary; the above shows that such measures on these groups have infinite entropy.
Similar content being viewed by others
References
Avez, A.: Entropie des groupes de type fini. C. R. Acad. Sci. Paris, Sér. I, Math. 275A, 1363–1366 (1972)
Avez, A.: Théorème de Choquet-Deny pour les groupes à croissance non exponentielle. C. R. Acad. Sci. Paris, Sér. I, Math. 279, 25–28 (1974)
Avez, A.: Croissance des groupes de type fini et fonctions harmoniques. Théorie ergodique (Actes Journées Ergodiques, Rennes, 1973/1974), pp. 35–49. Lect. Notes Math. 532. Berlin: Springer 1976
Avez, A.: Harmonic functions on groups. Differential geometry and relativity, pp. 27–32. Math. Phys. Appl. Math., Vol. 3. Dordrecht: Reidel 1976
Azencott, R.: Espaces de Poisson des groupes localement compacts. Lect. Notes Math. Vol. 148. Berlin, New York: Springer 1970
Baumslag, G.: Subgroups of finitely presented metabelian groups. J. Aust. Math. Soc. 16, 98–110 (1973)
Benjamimi, Y.: Instability of the Liouville property for quasi-isometric graphs and manifolds of polynomial volume growth. J. Theor. Probab. 4, 631–637 (1991)
Birge, L., Raugi, A.: Fonctions harmoniques sur les groupes moyennables. C. R. Acad. Sci. Paris, Sér. I, Math. 278, 1287–1289 (1974)
Blackwell, D.: On transient Markov processes with a countable number of states and stationary transition probabilities. Ann. Math. Statist. 26, 654–658 (1955)
Choquet, G., Deny, J.: Sur lé quation de convolution μ=μ*σ. C. R. Acad. Sci. Paris, Sér. I, Math. 250, 799–801 (1960)
Dynkin, E.B., Maljutov, M.B.: Random walk on groups with a finite number of generators. Dokl. Akad. Nauk SSSR 137, 1042–1045 (1961)
Derriennic, Y.: Quelques applications du théoreme ergodique sous-additif. Astérisque 74, 183–201 (1980)
Dyubina, A.: Characteristics of random walks on wreath product of groups. Zapiski Semin. POMI 256, 31–37 (1999)
Erschler (Dyubina), A.: Drift and entropy growth for random walk on groups. Russ. Math. Surv. 56, 179–180 (2001)
Erschler, A.: Drift and entropy growth for random walks on groups. To appear in Ann. Probab.
Erschler, A.: Boundary behaviour for groups of subexponential growth. Preprint, January 2003
Guivarch, Y.: Marches aleatoires à pas markoviens. C. R. Acad. Sci. Paris, S.r. A-B 289, 541–543 (1979)
Dynkin, E.B., Maljutov, M.B.: Random walk on groups with a finite number of generators. Dokl. Akad. Nauk SSSR 137, 1042–1045 (1961)
Furstenberg, H.: A Poisson formula for semi-simple Lie groups. Ann. Math. 77, 335–386 (1963)
Furstenberg, H.: Boundary theory and stochastic processes on homogeneous spaces. Proc. Symp. Pure Math. 26, 193–229 (1974). Providence, R.I.: Am. Math. Soc.
Gromov, M.: Metric structures for Riemannian and non-Riemannians paces. Progr. in Mathematics, 152. Boston, MA: Birkhäuser 1999
Kaimanovich, V.A.: Brownian motion and harmonic function on covering manifolds. An entropic approach. Dokl. Akad. Nauk SSSR 288, 1045–1049 (1986)
Kaimanovich, V.A.: Poisson Boundaries of random walks on discrete solvable groups. Probability measures on groups, X (Oberwolfach, 1990), 205–238. New York: Plenum 1991
Kaimanovich, V.A.: Discretisation of bounded harmonic functions on Riemannian manifolds and entropy. Potential theory (Nagoya, 1990), 213–223. Berlin: de Gruyter 1992
Kaimanovich, V.A., Vershik, A.M.: Random walks on groups: boundary, entropy, uniform distribution. Soviet. Math. Dokl. 20, 1170–1173 (1979)
Kaimanovich, V.A., Vershik, A.M.: Random walks on discrete groups: boundary and entropy. Ann. Probab. 11, 457–490 (1983)
Kesten, H.: Full Banach mean values on countable groups. Math. Scand. 7, 146–156 (1959)
Kropholler, P.H.: On finitely generated soluble groups with no large wreath product sections. Proc. London Math. Soc. (3) 49, 155–169 (1984)
Lyons, T.: Instability of Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains. J. Differ. Geom. 26, 33–66 (1987)
Lyons, T., Sullivan, D.: Fuction theory, random paths and covering spaces. J. Differ. Geom. 19, 299–323 (1984)
Margulis, G.A.: Positive harmonic functions on nilpotent groups. Dokl. Akad. Nauk SSSR 166, 1054–1057, translated as Soviet Math. Dokl. 7, 241–244 (1966)
Milnor, J.: Growth of finitely generated solvable groups. J. Differ. Geom. 2, 447–449 (1968)
Rohklin, V.A.: Lectures on entropic theory of transformations with invariant measure. Russ. Math. Surv. 22, 3–56 (1967)
Rosenblatt, J.: Ergodic and mixing random walks on locally compact groups. Math. Ann. 257, 31–42 (1981)
Spitzer, F.: Principles of random walk. Princeton: Van Nostrand 1964
Varopoulos, N.Th.: Théorie du potentiel sur des groupes et des variétés. C. R. Acad. Sci. Paris, Sér. I, Math. 302, 203–205 (1986)
Vershik, A.M.: Dynamical theory of growth of groups: entropy, boundaries, examples. Russ. Math. Surv. 55, 59–128 (2000)
Wolf, J.A.: Growth of finitely generated solvable groups and curvature of Riemannian manifolds. J. Differ. Geom. 2, 421–446 (1968)
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (1991)
60B15, 60J50, 28D20, 20P05, 43A07, 60J65, 43A85, 20f16
Rights and permissions
About this article
Cite this article
Erschler, A. Liouville property for groups and manifolds. Invent. math. 155, 55–80 (2004). https://doi.org/10.1007/s00222-003-0314-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-003-0314-7