Inventiones mathematicae

, Volume 206, Issue 3, pp 837–867 | Cite as

Extensions of amenable groups by recurrent groupoids

  • Kate Juschenko
  • Volodymyr Nekrashevych
  • Mikael de la Salle
Article

Abstract

We show that the amenability of a group acting by homeomorphisms can be deduced from a certain local property of the action and recurrency of the orbital Schreier graphs. This applies to a wide class of groups, the amenability of which was an open problem, as well as unifies many known examples to one general proof. In particular, this includes Grigorchuk’s group, Basilica group, group associated to Fibonacci tiling, the topological full groups of Cantor minimal systems, groups acting on rooted trees by bounded automorphisms, groups generated by finite automata of linear activity growth, and groups naturally appearing in holomorphic dynamics.

Mathematics Subject Classification

20F69 20L05 

Notes

Acknowledgments

We thank Omer Angel, Laurent Bartholdi, Rostislav Grigorchuk and Pierre de la Harpe for very useful and numerous comments on a previous version of this paper. In a preliminary version of the paper our results were not stated in terms of random walks but rather in term of the validity of certain inequalities. U. Bader, B. Hua and A. Valette suggested to look at a connection with recurrence of random walks and B. Hua pointed out [49, Theorem 3.24] to us. We thank them for this very fruitful suggestion and for other comments which are valuable to us. The research of M. de la Salle was supported by grants GAMME, NEUMANN and OSQPI of the Agence Nationale de la Recherche.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Kate Juschenko
    • 1
  • Volodymyr Nekrashevych
    • 2
  • Mikael de la Salle
    • 3
  1. 1.Department of MathematicsNorthwestern UniversityEvanstonUSA
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA
  3. 3.UMPA UMR CNRS 5669, ENS de Lyon, Université de LyonLyon Cedex 07France

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