Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Random walks on abelian-by-cyclic groups

Authors: Christophe Pittet and Laurent Saloff-Coste
Journal: Proc. Amer. Math. Soc. 131 (2003), 1071-1079
MSC (2000): Primary 20F69, 82B41, 60B99, 20F16
Published electronically: September 5, 2002
MathSciNet review: 1948097
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We describe the large time asymptotic behaviors of the probabilities $p_{2t}(e,e)$ of return to the origin associated to finite symmetric generating sets of abelian-by-cyclic groups. We characterize the different asymptotic behaviors by simple algebraic properties of the groups.

References [Enhancements On Off] (What's this?)

  • 1. R. Bieri and R. Strebel, Almost finitely presented soluble groups, Comment. Math. Helv. 53 (1978), no. 2, 258-278. MR 58:16890
  • 2. Th. Coulhon, A. Grigor'yan, and Ch. Pittet, A geometric approach to on-diagonal heat kernels lower bounds on groups, Annales de l'Institut Fourier 51 (2001), 1763-1827. (2002).
  • 3. P. de la Harpe, Topics on geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, 2000. MR 2001i:20081
  • 4. B. Farb and L. Mosher, A rigidity theorem for the solvable Baumslag-Solitar groups. With an appendix by Daryl Cooper, Invent. Math. 131 (1998), no. 2, 419-451. MR 99b:57003
  • 5. -, On the asymptotic geometry of abelian-by-cyclic groups, Acta Math. 184 (2000), no. 2, 145-202. MR 2001e:20035
  • 6. Ph. Hall, On the finiteness of certain solvable groups, Proc. London Math. Soc. 9 (1959), no. 3, 595-622. MR 22:1618
  • 7. I. Kaplansky, Infinite abelian groups, Ann Arbor The University of Michigan Press, 1969. MR 38:2208
  • 8. H. Kesten, Full banach mean values on countable groups, Math. Scand. 7 (1959), 146-156. MR 22:2911
  • 9. Ch. Pittet and L. Saloff-Coste, On random walks on wreath products, The Annals of Probability 30 (2002), no. 2, 948-997.
  • 10. -, Random walk and isoperimetry on discrete subgroups of Lie groups, Random walks and discrete potential theory, Cortona (M. Picardello and W. Woess, eds.), Walter de Gruyter, 1997, pp. 306-319. CMP 2001:06
  • 11. -, Amenable groups, isoperimeric profile and random walks, Geometric Group Theory Down Under (J. Cossey, Ch. F. Miller III, W. Neumann, and M. Shapiro, eds.), Walter de Gruyter, 1999, pp. 293-316. MR 2001d:20041
  • 12. -, On the stability of the behavior of random walks on groups, The Journal of Geometric Analysis 10 (2001), 701-726. CMP 2001:09
  • 13. W. Thurston, Three-dimensional geometry and topology, Princton Mathematical Series, 35, vol. 1, Princton University Press, 1997. MR 97m:57016
  • 14. N. Th. Varopoulos, Random walks on soluble groups, Bull. Sc. math. 2ème série 107 (1983), 337-344. MR 85e:60076
  • 15. N. Th. Varopoulos, Th. Coulhon, and L. Saloff-Coste, Analysis and geometry on groups, Cambridge Tracts in Math., vol. 100, Cambridge University Press, 1992. MR 95f:43008
  • 16. W. Woess, Random walks on infinite graphs and groups a survey on selected topics, Bull. London Math. Soc. 26 (1994), no. 1, 1-60. MR 94i:60081

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20F69, 82B41, 60B99, 20F16

Retrieve articles in all journals with MSC (2000): 20F69, 82B41, 60B99, 20F16

Additional Information

Christophe Pittet
Affiliation: Laboratoire Emile Picard, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France

Laurent Saloff-Coste
Affiliation: Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, New York 14853-4201

Keywords: Random walk, heat kernel decay, asymptotic invariants of infinite groups, metabelian groups
Received by editor(s): August 6, 2001
Received by editor(s) in revised form: November 19, 2001
Published electronically: September 5, 2002
Additional Notes: The first author was supported by a Delegation CNRS at UMR 5580
The second author was supported by NSF grant DMS-9802855
Communicated by: Jozef Dodziuk
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society