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 countable set X, a Følner sequence for the action is a sequence of finite subsets of X which exhaust X and which "don't move too much" when acted on by any group element. Precisely, Explanation of the notation used above: Thus, what this definition says is that for any group element g, the proportion of elements of F_i\ that are moved away by g goes to 0 as i gets large. In the setting of a locally compact group acting on a measure space (X,\mu) there is a more general definition. Instead of being finite, the sets are required to have finite, non-zero measure, and so the Følner requirement will be that analogously to the discrete case. The standard case is that of the group acting on itself by left translation, in which case the measure in question is normally assumed to be the Haar measure.

Examples

Proof of amenability

We have a group G and a Følner sequence F_i, and we need to define a measure \mu on G, which philosophically speaking says how much of G any subset A takes up. The natural definition that uses our Følner sequence would be 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: It turns out ultralimits have all the properties we need. Namely,