Martin boundary covers Floyd boundary

Abstract

For a random walk on a finitely generated group G we obtain a generalization of a classical inequality of Ancona. We deduce as a corollary that the identity map on G extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of conical points being singletons. This provides new results for Martin compactifications of relatively hyperbolic groups.

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Notes

  1. 1.

    The proof of the first part for hyperbolic groups is folklore and follows from different sources (e.g. [25, 26, 40]); a complete proof can be found in [35, Proposition A1, Appendix].

  2. 2.

    if T contains at most two points then \(\Theta ^3(T)=\emptyset \) and the action is convergence by definition.

  3. 3.

    In most cases the function f needs to only be defined on \(\mathbb N\cup \{0\}\), to cover all cases we consider it on \(\mathbb R_{\geqslant 0}\) .

  4. 4.

    We thank Wolfgang Woess for indicating to us this example.

  5. 5.

    In [19] this statement is formally stated for the Bowditch boundary but the proof equally works on the Floyd boundary as the only tool which is used is the Karlsson lemma (see Introduction).

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Acknowledgements

I.G. was partially supported by NSF grant DMS-1401875 and ERC advanced grant ‘Moduli’ of Prof. Ursula Hamenstädt. I.G., V.G. and L.P. are thankful to the Hausdorff center and to the Max-Planck Institut in Bonn for their research stay in 2016 when they started to work on this paper. V.G. and L.P. are grateful to the LABEX CEMPI in Lille for a partial support; they were also partly supported by MATH-AmSud (code 18-MATH-08) and by the Simons grant of L.P. at the CRM Institute of Montreal. W.Y. is supported by the National Natural Science Foundation of China (No. 11771022). The authors are deeply grateful to the referee for numerous remarks and suggestions which certainly ameliorated the paper.

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Correspondence to Leonid Potyagailo.

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Gekhtman, I., Gerasimov, V., Potyagailo, L. et al. Martin boundary covers Floyd boundary. Invent. math. 223, 759–809 (2021). https://doi.org/10.1007/s00222-020-01015-z

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Mathematics Subject Classification

  • Primary 20F65
  • 20F67
  • Secondary 57M07
  • 22D05