Skip to main content
SpringerLink
Log in

Poisson boundary for finitely generated groups of rational affinities

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

The group Aff(ℚ) of affine transformations with rational coefficients acts naturally not only on the real line ℝ, but also on the p-adic fields ℚp. The aim of this note is to show that all these actions are necessary and sufficient to represent bounded μ-harmonic functions for a probability measure μ on Aff(ℚ) that is supported by a finitely generated subgroup, that is, to describe the Poisson boundary.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Avez, “Théorème de Choquet-Deny pour les groupes à croissance non exponentielle,” C. R. Acad. Sci. Paris Sér. A, 279, 25–28 (1974).

    MATH  MathSciNet  Google Scholar 

  2. M. Babillot, P. Bougerol, and L. Elie, “The random difference equation X n = A n X n−1 + B n in the critical case,” Ann. Probab., 25, No. 1, 478–493 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  3. W. Ballmann and F. Ledrappier, “The Poisson boundary for rank one manifolds and their cocompact lattices,” Forum Math., 6, No. 3, 301–313 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  4. P. Bougerol and N. Picard, “Strict stationarity of generalized autoregressive processes,” Ann. Probab., 20, No. 4, 1714–1730 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  5. S. Brofferio, “Speed of locally contractive stochastic systems,” Ann. Probab., 31, No. 4, 2040–2067 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  6. D. I. Cartwright, V. A. Kaimanovich, and W. Woess, “Random walks on the affine group of local fields and of homogeneous trees,” Ann. Inst. Fourier, 44, No. 4, 1243–1288 (1994).

    MATH  MathSciNet  Google Scholar 

  7. Y. Derriennic, “Quelques applications du théorème ergodique sous-additif,” in: Conf. on Random Walks, Kleebach, 1979, French, Astérisque, 74, Soc. Math. France, Paris (1980), pp. 183–201.

    Google Scholar 

  8. L. Élie, “Comportement asymptotique du noyau potentiel sur les groupes de Lie,” Ann. Sci. École Norm. Sup. (4), 15, No. 2, 257–364 (1982).

    MATH  Google Scholar 

  9. B. Farb and L. Mosher, “Quasi-isometric rigidity for the solvable Baumslag-Solitar groups, II,” Invent. Math., 137, No. 3, 613–649 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  10. H. Furstenberg, “Noncommuting random products,” Trans. Amer. Math. Soc., 108, 377–428 (1963).

    Article  MATH  MathSciNet  Google Scholar 

  11. V. A. Kaimanovich, “Poisson boundaries of random walks on discrete solvable groups,” in: Probability Measures on Groups, X, Oberwolfach, 1990, Plenum, New York (1991), pp. 205–238.

    Google Scholar 

  12. V. A. Kaimanovich, “The Poisson formula for groups with hyperbolic properties,” Ann. of Math. (2), 152, No. 3, 659–692 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  13. H. Kesten, “Random difference equations and renewal theory for products of random matrices,” Acta Math., 131, 207–248 (1973).

    Article  MATH  MathSciNet  Google Scholar 

  14. J. Milnor, “A note on curvature and fundamental group,” J. Differ. Geom., 2, 1–7 (1968).

    MATH  MathSciNet  Google Scholar 

  15. A. M. Vershik and V. A. Kaimanovich, “Random walks on groups: boundary, entropy, uniform distribution,” Dokl. Akad. Nauk SSSR, 249, No. 1, 15–18 (1979).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik C, Technische Universität, Graz, Austria

    S. Brofferio

Authors
  1. S. Brofferio
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Brofferio.

Additional information

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 50, Functional Analysis, 2007.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brofferio, S. Poisson boundary for finitely generated groups of rational affinities. J Math Sci 156, 1–10 (2009). https://doi.org/10.1007/s10958-008-9253-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-008-9253-6

Keywords

Search

Navigation