Topological groups and recurrence of quasi transitive graphs

  • Wolfgang Woess


We present all steps which are necessary in order to classify all locally finite, infinite graphs which carry a quasi transitive random walk that is recurrent. Some new and/or simpler proofs are given. Most of them rely on the fact that autmomorphism groups of locally finite graphs are locally compact with respect to the topology of pointwise convergence—this allows the use of integration on these groups.


Random Walk Topological Group Cayley Graph Isoperimetric Inequality Polynomial Growth 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [An]Ancona, A.,Théorie du potentiel sur les graphes et les varietes, École d’Été de Probabilités de Saint-Flour XVIII – 1988 (ed. P. L. Hennequin), Springer Lect. Notes in Math.1427, Berlin, 1990, 4–112.Google Scholar
  2. [Ba]Bass, H.,The degree of growth of finitely generated nilpotent groups, Proc. London Math. Soc.25 (1972), 603–614.CrossRefMathSciNetMATHGoogle Scholar
  3. [B-L-P]Baldi, P., Lohué, N. andPeyrière, J.,Sur la classification des groupes récurrents, C. R. Acad. Sc. Paris, Série A,285 (1977), 1103–1104.MATHGoogle Scholar
  4. [C-K-S]Carlen, E., Kusuoka S. andStroock, D.,Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré (Probab. Stat.)26 (1990), 245–287.MathSciNetGoogle Scholar
  5. [Ch]Chen, M. F.,Comparison theorems for Green functions of Markov chains, Chinese Ann. of Math.3 (1991), 237–242.Google Scholar
  6. [C-S1]Coulhon, T. andSaloff-Coste, L.,Puissances d’un opérateur regularisant, Ann. Inst. H. Poincaré (Probab. Stat.)26 (1990), 419–436.MathSciNetMATHGoogle Scholar
  7. [C-S2]Coulhon, T. andSaloff-Coste, L.,Isopérimétrie pour les groupes et ls variétés, Rev. Mt. Iberoamericana9 (1993), 293–314.MathSciNetMATHGoogle Scholar
  8. [D-S]Doyle, P. G. andSnell, J.L.,Random Walks and Electric Networks, The Carus Math. Monographs22, Math. Association of America, 1984.Google Scholar
  9. [Ge]Gerl, P.,Rekurrente und transiente Bäume, in: Séminaire Lotharingien de Combinatoire (IRMA Strasbourg)10 (1984), 80–87.Google Scholar
  10. [Gr]Gromov, M.,Groups of polynomial growth and expanding maps, Publ. Math. I. H. E. S.53 (1981), 53–73.MathSciNetMATHGoogle Scholar
  11. [G-K-R]Guivarc'h, Y., Keane, M. andRoynette, B. Marches Aléatoires sur les groupes de Lie, Lect. Notes in Math.624 Springer, Berlin, 1977.MATHGoogle Scholar
  12. [K1]Kanai, M.,Rough isometries and combinatorial approximations of geometries of non-compact Riemannian manifolds, J. Math. Soc. Japan37 (1985), 391–413.MathSciNetMATHCrossRefGoogle Scholar
  13. [K2]Kanai, M.,Rough isometries and the parabolicity of Riemannian manifolds, J. Math. Soc. Japan38, (1986), 227–238.MathSciNetMATHGoogle Scholar
  14. [K-S-K]Kemeny, J. G., Snell, J. L. andKnapp A. W.,Denumerable Markov Chains (2nd edition), Springer, New York, 1976.MATHGoogle Scholar
  15. [Ke]Kesten, H.,The Martin boundary for recurrent random walks on countable groups, Proc. 5th Berkeley Sympos. on Math. Statistics and Probability, vol. 2, Univ. of California Press, Berkeley, 1967, 51–74.Google Scholar
  16. [Lo]Losert, V.,On the structure of groups with polynomial growth, Math. Zeitschr.195 (1987), 109–117.CrossRefMathSciNetMATHGoogle Scholar
  17. [Po]Pólya, G.,Über eine Aufgabe der Wahrscheinlichkeitstheorie betreffend die Irrfahrt im Straßennetz, Math. Ann.84 (1921), 149–160.CrossRefMathSciNetMATHGoogle Scholar
  18. [Sb]Sabidussi, G.,Vertex-transitive graphs, Monatsh. Math68 (1964), 427–438.CrossRefMathSciNetGoogle Scholar
  19. [SC]Saloff-Coste, L.,Isoperimetric inequalities and decay of iterated kernels for almost-transitive Markov chains, in print, Probability, Combinatorics, and Computing.Google Scholar
  20. [Sl]Salvatori, M.,On the norms of group-invariant transition operators on graphs, J. Theoret. Probab.5 (1991), 563–576.CrossRefMathSciNetGoogle Scholar
  21. [S-W]Soardi, P. M. andWoess, W.,Amenability, unimodularity, and the spectral radius of random walks on infinite graphs, Math. Zeitschr.205 (1990), 471–486.CrossRefMathSciNetMATHGoogle Scholar
  22. [S-Y]Soardi, P. M. andYamasaki M.,Classification of infinite networks and its applications, Circuits, Systems, and Signal Processing12 (1993), 133–149.CrossRefMATHGoogle Scholar
  23. [T1]Thomassen, C.,Isoperimetric inequalities and transient random walks on graphs, Ann. Probab.20 1592–1600.Google Scholar
  24. [T2]Thomassen, C.,Trees, ends, and transience, Harmonic Analysis and Discrete Potential Theory (ed. M. A. Picardello), Plenum, New York, 1992, 259–266.Google Scholar
  25. [Tr]Trofimov, V. I.,Graphs with polynomial growth, Math. USSR Sbornik51 (1985), 405–417.CrossRefMATHGoogle Scholar
  26. [V1]Varopoulos, N. Th.,Isoperimetric inequalities and Markov chains, J. Funct. Analysis63 (1985), 215–239.CrossRefMathSciNetMATHGoogle Scholar
  27. [V2]Varopoulos, N. Th.,Théorie du potentiel sur des groupes et des variétés, C. R. Acad. Sc. Paris, Série I,302 (1986), 203–205.MathSciNetMATHGoogle Scholar
  28. [V3]Varopoulos, N. Th.,Convolution powers on locally compact groups, Bull. Sc. Math.111 (1987), 333–342.MathSciNetMATHGoogle Scholar
  29. [V-S-C]Varopoulos, N. Th., Saloff-Coste, L. andCoulhon, T., Analysis and Geometry on Groups, Cambridge Univ. Press, 1993.Google Scholar
  30. [W1]Woess, W.,Boundaries of random walks on graphs and groups with infinitely many ends, Israel J. Math.68 (1989), 271–301.CrossRefMathSciNetMATHGoogle Scholar
  31. [W2]Woess, W.,Topological groups and infinite graphs, Discrete Math.95 (1991), 373–384.CrossRefMathSciNetMATHGoogle Scholar
  32. [W3]Woess, W.,Random walks on infinite graphs and groups: a survey on selected topics, Bull. London Math. Soc.26 (1994), 1–60.CrossRefMathSciNetMATHGoogle Scholar
  33. [Y1]Yamasaki, M.,Parabolic and hyperbolic infinite networks, Hiroshima Math. J.7 (1977), 135–146.MathSciNetMATHGoogle Scholar
  34. [Y2]Yamasaki, M.,Discrete potentials on an infinite network, Mem. Fac. Sci. Shimane Univ.13 (1979), 31–44.MathSciNetMATHGoogle Scholar

Copyright information

© Birkhäuser-Verlag 1994

Authors and Affiliations

  • Wolfgang Woess
    • 1
  1. 1.Dipartimento di MatematicaUniversità di MilanoMilanoItalia

Personalised recommendations