Liouville property for groups and manifolds

Inventiones Mathematicae

Volumn 155, Issue 1, 2004, Pages 55-80

  Erschler, Anna ^a,b  

^a STEKLOV MATHEMATICAL INSTITUTE   (Russian Federation)

^b INSTITUT DES HAUTES ÉTUDES SCIENTIFIQUES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0742305918     PISSN: 00209910     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00222-003-0314-7     Document Type: Article

References (39)

1. Avez, A.: Entropie des groupes de type fini. C. R. Acad. Sci. Paris, Sér. I, Math. 275A, 1363-1366 (1972)
2. 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)
3. 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
4. Avez, A.: Harmonic functions on groups. Differential geometry and relativity, pp. 27-32. Math. Phys. Appl. Math., Vol. 3. Dordrecht: Reidel 1976
5. Azencott, R.: Espaces de Poisson des groupes localement compacts. Lect. Notes Math. Vol. 148. Berlin, New York: Springer 1970
6. Baumslag, G.: Subgroups of finitely presented metabelian groups. J. Aust. Math. Soc. 16, 98-110 (1973)
7. Benjamimi, Y: Instability of the Liouville property for quasi-isometric graphs and manifolds of polynomial volume growth. J. Theor. Probab. 4, 631-637 (1991)
8. Birge, L., Raugi, A.: Fonctions harmoniques sur les groupes moyennables. C. R. Acad. Sci. Paris, Sér. I, Math. 278, 1287-1289 (1974)
9. Blackwell, D.: On transient Markov processes with a countable number of states and stationary transition probabilities. Ann. Math. Statist. 26, 654-658 (1955)
10. Choquet, G., Deny, J.: Sur lé quation de convolution μ = μ* σ. C. R. Acad. Sci. Paris, Sér. I, Math. 250, 799-801 (1960)
11. Dynkin, E.B., Maljutov, M.B.: Random walk on groups with a finite number of generators. Dokl. Akad. Nauk SSSR 137, 1042-1045 (1961)
12. Derriennic, Y: Quelques applications du théoreme ergodique sous-additif. Astérisque 74, 183-201 (1980)
13. Dyubina, A.: Characteristics of random walks on wreath product of groups. Zapiski Semin. POMI 256, 31-37 (1999)
14. Erschler (Dyubina), A.: Drift and entropy growth for random walk on groups. Russ. Math. Surv. 56, 179-180 (2001)
15. Erschler, A.: Drift and entropy growth for random walks on groups. To appear in Ann. Probab.
16. Erschler, A.: Boundary behaviour for groups of subexponential growth. Preprint, January 2003
17. Guivarch, Y.: Marches aleatoires à pas markoviens. C. R. Acad. Sci. Paris, S.r. A-B 289, 541-543 (1979)
18. Dynkin, E.B., Maljutov, M.B.: Random walk on groups with a finite number of generators. Dokl. Akad. Nauk SSSR 137, 1042-1045 (1961)
19. Furstenberg, H.: A Poisson formula for semi-simple Lie groups. Ann. Math. 77, 335386 (1963)
20. Furstenberg, H.: Boundary theory and stochastic processes on homogeneous spaces. Proc. Symp. Pure Math. 26, 193-229 (1974). Providence, R.I.: Am. Math. Soc.
21. Gromov, M.: Metric structures for Riemannian and non-Riemannians paces. Progr. in Mathematics, 152. Boston, MA: Birkhäuser 1999
22. Kaimanovich, V.A.: Brownian motion and harmonic function on covering manifolds. An entropie approach. Dokl. Akad. Nauk SSSR 288, 1045-1049 (1986)
23. 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
24. Kaimanovich, V.A.: Discretisation of bounded harmonic functions on Riemannian manifolds and entropy. Potential theory (Nagoya, 1990), 213-223. Berlin: de Gruyter 1992
25. Kaimanovich, V.A., Vershik, A.M.: Random walks on groups: boundary, entropy, uniform distribution. Soviet. Math. Dokl. 20, 1170-1173 (1979)
26. Kaimanovich, V.A., Vershik, A.M.: Random walks on discrete groups: boundary and entropy. Ann. Probab. 11, 457-490 (1983)
27. Kesten, H.: Full Banach mean values on countable groups. Math. Scand. 7, 146-156 (1959)
28. Kropholler, P.H.: On finitely generated soluble groups with no large wreath product sections. Proc. London Math. Soc. (3) 49, 155-169 (1984)
29. Lyons, T.: Instability of Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains. J. Differ. Geom. 26, 33-66 (1987)
30. Lyons, T., Sullivan, D.: Fuction theory, random paths and covering spaces. J. Differ. Geom. 19, 299-323 (1984)
31. 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)
32. 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)
33. Milnor, J.: Growth of finitely generated solvable groups. J. Differ. Geom. 2, 447-449 (1968)
34. Rohklin, V.A.: Lectures on entropic theory of transformations with invariant measure. Russ. Math. Surv. 22, 3-56 (1967)
35. Rosenblatt, J.: Ergodic and mixing random walks on locally compact groups. Math. Ann. 257, 31-42 (1981)
36. Spitzer, F.: Principles of random walk. Princeton: Van Nostrand 1964
37. Varopoulos, N.Th.: Théorie du potentiel sur des groupes et des variétés. C. R. Acad. Sci. Paris, Sér. I, Math. 302, 203-205 (1986)
38. Vershik, A.M.: Dynamical theory of growth of groups: entropy, boundaries, examples. Russ. Math. Surv. 55, 59-128 (2000)
39. Wolf, J.A.: Growth of finitely generated solvable groups and curvature of Riemannian manifolds. J. Differ. Geom. 2, 421-446 (1968)