TY - JOUR
T1 - Extensions of amenable groups by recurrent groupoids
AU - Juschenko, Kate
AU - Nekrashevych, Volodymyr
AU - de la Salle, Mikael
N1 - Funding Information:
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 [, 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.
Funding Information:
V. Nekrashevych was supported by NSF Grant DMS1006280. M. de la Salle was supported by ANR Grants OSQPI and NEUMANN.
Publisher Copyright:
© 2016, Springer-Verlag Berlin Heidelberg.
PY - 2016/12/1
Y1 - 2016/12/1
N2 - 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.
AB - 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.
KW - 20F69
KW - 20L05
UR - http://www.scopus.com/inward/record.url?scp=84968571725&partnerID=8YFLogxK
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U2 - 10.1007/s00222-016-0664-6
DO - 10.1007/s00222-016-0664-6
M3 - Article
AN - SCOPUS:84968571725
SN - 0020-9910
VL - 206
SP - 837
EP - 867
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
IS - 3
ER -