A discrete group is amenable when it admits a left-invariant finitely additive probability measure on all subsets of . Equivalently, it admits an invariant mean on the bounded real-valued functions on .
An invariant mean on a group is a positive linear functional with and for every . Applied to indicator functions, it is a left-invariant finitely additive probability measure on all subsets of .
For a group acting on a set , subsets are -equidecomposable when there are finite partitions and and elements with for every .
A -set is paradoxical when it contains two disjoint subsets each -equidecomposable with . An invariant finitely additive probability measure rules out such a decomposition, because it would assign each copy the full measure of .
The free group is not an amenable group. Partitioning reduced words by their first letters gives two translated decompositions of whose invariance equations add to , contradicting the existence of an invariant mean.
Every subgroup of an amenable group is amenable. One proof restricts an invariant mean after extending bounded functions from the subgroup to the whole group along a set of coset representatives. Consequently, a group containing a nonamenable subgroup is nonamenable.
Every abelian group is amenable, and amenability is preserved by finite extensions. Hence every group with an abelian finite-index subgroup is an amenable group.
A finitely generated group with finite generating set satisfies the Følner condition when, for every , there is a nonempty finite subset such thatThe condition is independent of the finite generating set.
Normalized counting measures on a Følner sequence have a convergent subnet in the compact product . The limit is a left-invariant finitely additive probability measure, so every group satisfying the Følner condition is an amenable group.
For nontrivial groups , the free product is amenable exactly in the exceptional case . In that case it is the infinite dihedral group, hence a virtually cyclic group and amenable. In every other case the normal form theorem for a free product and the ping-pong lemma produce a subgroup isomorphic to ; the fact that a subgroup of an amenable group is amenable then obstructs amenability.
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An **amenable group** is a type of mathematical structure studied in the field of group theory, specifically in the study of topological groups and functional analysis. The concept of amenability is related to the ability of a group to have a certain type of "invariance" property under averaging processes. A group \( G \) is called **amenable** if it has a left-invariant mean.