Følner sequence

In mathematics, a Følner sequence for a group is a sequence of sets satisfying a particular condition. If a group has a Følner sequence with respect to its action on itself, the group is amenable. A more general notion of Følner nets can be defined analogously, and is suited for the study of uncountable groups. Følner sequences are named for Erling Følner.

Definition

Given a group $$G that acts on a set $$X, a Følner sequence for the action is a sequence of finite subsets $$F_1, F_2, dots of $$X which exhaust $$X and which "don't move too much" when acted on by any group element. Precisely,:For every $$xin X, there exists some $$i such that $$x in F_j for all $$j > i, and:$$lim_{i oinfty}frac.Of course, this limit doesn't necessarily exist. To overcome this technicality, we take an ultrafilter $$U on the natural numbers that contains intervals $$ [n,infty). Then we use an ultralimit instead of the regular limit::$$mu(A)=U{ extrm-}lim ight| = left| o0:by the Følner sequence definition.

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