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.

References

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