Random walks on abelian-by-cyclic groups
HTML articles powered by AMS MathViewer
- by Christophe Pittet and Laurent Saloff-Coste PDF
- Proc. Amer. Math. Soc. 131 (2003), 1071-1079 Request permission
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
- Robert Bieri and Ralph Strebel, Almost finitely presented soluble groups, Comment. Math. Helv. 53 (1978), no. 2, 258–278. MR 498863, DOI 10.1007/BF02566077
- 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. http://geometry.ma.ic.ac.uk/~grigor/pubs.htm (2002).
- Pierre de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. MR 1786869
- Benson Farb and Lee Mosher, A rigidity theorem for the solvable Baumslag-Solitar groups, Invent. Math. 131 (1998), no. 2, 419–451. With an appendix by Daryl Cooper. MR 1608595, DOI 10.1007/s002220050210
- Benson Farb and Lee Mosher, On the asymptotic geometry of abelian-by-cyclic groups, Acta Math. 184 (2000), no. 2, 145–202. MR 1768110, DOI 10.1007/BF02392628
- P. Hall, On the finiteness of certain soluble groups, Proc. London Math. Soc. (3) 9 (1959), 595–622. MR 110750, DOI 10.1112/plms/s3-9.4.595
- Irving Kaplansky, Infinite abelian groups, Revised edition, University of Michigan Press, Ann Arbor, Mich., 1969. MR 0233887
- Harry Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 146–156. MR 112053, DOI 10.7146/math.scand.a-10568
- Ch. Pittet and L. Saloff-Coste, On random walks on wreath products, The Annals of Probability 30 (2002), no. 2, 948–997.
- —, 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.
- Christophe Pittet and Laurent Saloff-Coste, Amenable groups, isoperimetric profiles and random walks, Geometric group theory down under (Canberra, 1996) de Gruyter, Berlin, 1999, pp. 293–316. MR 1714851
- —, On the stability of the behavior of random walks on groups, The Journal of Geometric Analysis 10 (2001), 701–726.
- William P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy. MR 1435975
- Nicholas Th. Varopoulos, Random walks on soluble groups, Bull. Sci. Math. (2) 107 (1983), no. 4, 337–344 (English, with French summary). MR 732356
- N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and geometry on groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press, Cambridge, 1992. MR 1218884
- Wolfgang Woess, Random walks on infinite graphs and groups—a survey on selected topics, Bull. London Math. Soc. 26 (1994), no. 1, 1–60. MR 1246471, DOI 10.1112/blms/26.1.1
Additional Information
- Christophe Pittet
- Affiliation: Laboratoire Emile Picard, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
- Email: pittet@picard.ups-tlse.fr
- Laurent Saloff-Coste
- Affiliation: Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, New York 14853-4201
- MR Author ID: 153585
- Email: lsc@math.cornell.edu
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 1071-1079
- MSC (2000): Primary 20F69, 82B41, 60B99, 20F16
- DOI: https://doi.org/10.1090/S0002-9939-02-06674-1
- MathSciNet review: 1948097