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Amenability and the Liouville property. (English) Zbl 1080.43003

The notion of amenability for groups is, from the analytic point of view, the most natural generalization of finiteness or compactness. Amenable groups are those which admit an invariant mean (rather than an invariant probability measure, which is the case for compact groups). One of the main applications of amenability is the fixed point property for affine actions of amenable groups on compact spaces. It turns out that non-amenable groups may still have actions which look like actions of amenable groups. This observation led to introduce the notion of an amenable action. Another generalization is the notion of amenability for equivalence relations and foliations. All these objects can be considered as measured groupoids, and the notion of amenability in each particular case is a specialization of the general notion of an amenable measured groupoid.
In this paper, the author presents a new approach to the amenability of groupoids based on Markov operators. He introduces the notion of an invariant Markov operator on a groupoid and shows that the Liouville property for such an operator implies the amenability of the groupoid. Moreover, the groupoid action on the Poisson boundary of any invariant operator is always amenable. This approach subsumes as particular cases numerous earlier results on the amenability for groups, actions, equivalence relations and foliations.

MSC:

43A07 Means on groups, semigroups, etc.; amenable groups
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