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Measures invariant under actions of \(F_ 2\). (English) Zbl 0701.43001

Let G be a group, with identity element 1, acting on the set S equipped with a finitely additive probability measure \(\mu\) ; the latter is said to be G-invariant if \(\mu (gX)=\mu (X)\) whenever \(g\in G\) and \(X\subset S\). In case G is amenable, S admits such a G-invariant measure; if S admits a G-invariant measure and gs\(\neq s\) for all \(s\in S\), \(g\in G\), \(g\neq 1\), then G is amenable.
The importance of the main result is stressed by the fact that the free group \(F_ 2\) on two generators constitutes the simplest nonamenable group: There exists a transitive action of \(F_ 2\) on a countably infinite set S such that \(\{\) \(s\in S:\) \(fs=s\}\) is finite whenever \(f\in F_ 2\), \(f\neq 1\), and S admits an \(F_ 2\)-invariant measure. The example may be adjusted in order that for every \(f\in F_ 2\) there exists \(s\in S\) such that \(fs=s\). More specifically, S may be any countable infinite amenable group.
The following result is also provided: For every infinite amenable group H there is a nonamenable group G acting faithfully and transitively on H such that every left-invariant measure on H is G-invariant.
Reviewer: J.-P.Pier

MSC:

43A05 Measures on groups and semigroups, etc.
20E05 Free nonabelian groups
43A07 Means on groups, semigroups, etc.; amenable groups
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