Israel Journal of Mathematics

, Volume 44, Issue 3, pp 221–242 | Cite as

Martin boundaries of random walks on Fuchsian groups

  • Caroline Series
Article

Abstract

Let Γ be a finitely generated non-elementary Fuchsian group, and let μ be a probability measure with finite support on Γ such that supp μ generates Γ as a semigroup. If Γ contains no parabolic elements we show that for all but a small number of co-compact Γ, the Martin boundaryM of the random walk on Γ with distribution μ can be identified with the limit set Λ of Γ. If Γ has cusps, we prove that Γ can be deformed into a group Γ', abstractly isomorphic to Γ, such thatM can be identified with Λ', the limit set of Γ'. Our method uses the identification of Λ with a certain set of infinite reduced words in the generators of Γ described in [15]. The harmonic measure ν (ν is the hitting distribution of random paths in Γ on Λ) is a Gibbs measure on this space of infinite words, and the Poisson boundary of Γ, μ can be identified with Λ, ν.

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Copyright information

© Hebrew University 1983

Authors and Affiliations

  • Caroline Series
    • 1
  1. 1.Mathematics InstituteUniversity of WarwickCoventryEngland

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