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Poisson–Furstenberg boundary and growth of groups

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Abstract

We study the Poisson–Furstenberg boundary of random walks on permutational wreath products. We give a sufficient condition for a group to admit a symmetric measure of finite first moment with non-trivial boundary, and show that this criterion is useful to establish exponential word growth of groups. We construct groups of exponential growth such that all finitely supported (not necessarily symmetric, possibly degenerate) random walks on these groups have trivial boundary. This gives a negative answer to a question of Kaimanovich and Vershik.

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Authors and Affiliations

  1. École Normale Supérieure, Paris, France

    Laurent Bartholdi & Anna Erschler

  2. Georg-August Universität zu Göttingen, Göttingen, France

    Laurent Bartholdi

Authors
  1. Laurent Bartholdi
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  2. Anna Erschler
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Correspondence to Laurent Bartholdi.

Additional information

The work is supported by the ERC starting Grant GA 257110 “RaWG”, the Courant Research Centre “Higher Order Structures” of the University of Göttingen, and the ANR Grant ANR-14-ACHN-0018-01.

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Bartholdi, L., Erschler, A. Poisson–Furstenberg boundary and growth of groups. Probab. Theory Relat. Fields 168, 347–372 (2017). https://doi.org/10.1007/s00440-016-0712-6

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  • DOI: https://doi.org/10.1007/s00440-016-0712-6

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