Extensions of amenable groups by recurrent groupoids

Abstract

We show that 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 covers amenability of a wide class 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, the full topological group of Cantor minimal system, groups acting on rooted trees by bounded automorphisms, groups generated by finite automata of linear activity growth, groups that naturally appear in holomorphic dynamics.

The last condition of the main thm was removed, the proofs are streamlined

Cited by in corpus (19)

• On Virtual Conjugacy of Generalized Odometers
• Extensive amenability and an application to interval exchanges
• Invariant means and the structure of inner amenable groups
• On spectra and spectral measures of Schreier and Cayley graphs
• Subgroup dynamics and $C^{\ast }$-simplicity of groups of homeomorphisms
• Isoperimetric profiles and random walks on some permutation wreath products
• The Liouville property for groups acting on rooted trees
• Ideal structure of the C*-algebra of Thompson group T
• On topological full groups of Z^d-actions
• Property FW, differentiable structures, and smoothability of singular actions
• Realizing residually finite groups as subgroups of branch groups
• Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces
• Amenability of quadratic automaton groups
• Amenability of Bounded Automata Groups on Infinite Alphabets
• Representing topological full groups in Steinberg algebras and C*-algebras
• Topological full groups of line-like minimal group actions are amenable
• Invariant means on Boolean inverse monoids
• Subshifts with slow complexity and simple groups with the Liouville property
• Constructing number field isomorphisms from *-isomorphisms of certain crossed product C*-algebras