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Cantor systems, piecewise translations and simple amenable groups
"... Abstract We provide the first examples of finitely generated simple groups that are amenable (and infinite). To this end, we prove that topological full groups of minimal systems are amenable. This follows from a general existence result on invariant states for piecewisetranslations of the integer ..."
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Abstract We provide the first examples of finitely generated simple groups that are amenable (and infinite). To this end, we prove that topological full groups of minimal systems are amenable. This follows from a general existence result on invariant states for piecewisetranslations of the integers. The states are obtained by constructing a suitable family of densities on the classical Bernoulli space.
Amenable actions, free products and a fixed point property
"... Abstract. We investigate the class of groups admitting an action on a set with an invariant mean. It turns out that many free products admit such an action. We give a complete characterisation of such free products in terms of a strong fixed point property. 1. ..."
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Abstract. We investigate the class of groups admitting an action on a set with an invariant mean. It turns out that many free products admit such an action. We give a complete characterisation of such free products in terms of a strong fixed point property. 1.
Amenable actions and almost invariant sets
 Proc. Amer. Math. Soc
"... Abstract. In this paper, we study the connections between properties of the action of a countable group Γ on a countable set X and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of Γ on MX, where M is a measure space. In parti ..."
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Abstract. In this paper, we study the connections between properties of the action of a countable group Γ on a countable set X and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of Γ on MX, where M is a measure space. In particular, we show that the action of Γ on X is amenable iff the shift Γ ↪→MX has almost invariant sets. 1.
Amenable actions of nonamenable groups
"... Since 1929 when von Neumann [vN29] introduced the notion of an invariant mean on a group (and more generally on a Gset) there is a permanent interest in the study of the phenomenon known as amenability. Amenable objects like groups, semigroups, algebras, graphs, metric spaces, operator algebras etc ..."
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Cited by 10 (1 self)
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Since 1929 when von Neumann [vN29] introduced the notion of an invariant mean on a group (and more generally on a Gset) there is a permanent interest in the study of the phenomenon known as amenability. Amenable objects like groups, semigroups, algebras, graphs, metric spaces, operator algebras etc. play an important role in different areas of mathematics. A big progress
Amenable, transitive and faithful actions of groups acting on trees
"... We study under which condition an amalgamated free product or an HNNextension over a finite subgroup admits an amenable, transitive and faithful action on an infinite countable set. We show that such an action exists if the initial groups admit an amenable and almost free action with infinite orbi ..."
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Cited by 2 (1 self)
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We study under which condition an amalgamated free product or an HNNextension over a finite subgroup admits an amenable, transitive and faithful action on an infinite countable set. We show that such an action exists if the initial groups admit an amenable and almost free action with infinite orbits (e.g. virtually free groups or infinite amenable groups). Our result relies on the Baire category Theorem. We extend the result to groups acting on trees.