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.

Definition

Given a group <math>G<math> that acts on a set <math>X<math>, a Følner sequence for the action is a sequence of finite subsets <math>F_1, F_2, \dots<math> of <math>X<math> that "don't move too much" when acted on by any group element. Precisely,

<math>\lim_{i\to\infty}\frac{|gF_i\,\triangle\,F_i|}{|F_i|} = 0<math> for all group elements <math>g<math> in <math>G<math>.

Explanation of the notation used above:

• <math>gF_i<math> is the set <math>F_i<math> acted on the left by <math>g<math>. It consists of elements of the form <math>gf<math> for all <math>f<math> in <math>F_i<math>.
• <math>\triangle<math> is the symmetric difference operator.
• <math>|A|<math> is the cardinality of a set <math>A<math>.

Thus, what this definition says is that for any group element <math>g<math>, the percent of elements of <math>F_i<math> that are moved away by <math>g<math> goes to 0 as <math>i<math> gets large.

Examples

• Any finite group <math>G<math> trivially has a Følner sequence <math>F_i=G<math> for each <math>i<math>.

• Consider the group of integers, acting on itself by addition. Let <math>F_i<math> consist of the integers between <math>-i<math> and <math>i<math>. Then <math>gF_i<math> consists of integers between <math>g-i<math> and <math>g+i<math>. The symmetric difference has size <math>2g<math>, while <math>F_i<math> has size <math>2i+1<math>, so the ratio is <math>2g/(2i+1)<math>, which goes to 0 as <math>i<math> gets large.

Proof of amenability

We have a group <math>G<math> and a Følner sequence <math>F_i<math>, and we need to define a measure <math>\mu<math> on <math>G<math>, which philosophically speaking says how much of <math>G<math> any subset <math>A<math> takes up. The natural definition that uses our Følner sequence would be

<math>\mu(A)=\lim_{i\to\infty}{|A\cap F_i|\over|F_i|}<math>.

Of course, this limit doesn't necessarily exist. To overcome this technicality, we take an ultrafilter <math>U<math> on the natural numbers that contains intervals <math>[n,\infty)<math>. Then we use an ultralimit instead of the regular limit:

<math>\mu(A)=U{\textrm-}\lim{|A\cap F_i|\over|F_i|}<math>.

It turns out ultralimits have all the properties we need. Namely,

1. <math>\mu<math> is a probability measure. That is, <math>\mu(G)=U\textrm{-}\lim1=1<math>, since the ultralimit coincides with the regular limit when it exists.
2. <math>\mu<math> is finitely additive. This is since ultralimits commute with addition just as regular limits do.
3. <math>\mu<math> is left invariant. This is since

   <math>\left|{|gA\cap F_i|\over|F_i|}-{|A\cap F_i|\over|F_i|}\right| = \left|{|A\cap g^{-1}F_i|\over|F_i|}-{|A\cap F_i|\over|F_i|}\right|<math>

   <math>\leq{|A\cap(g^{-1}F_i\,\triangle\,F_i)|\over|F_i|}\to0<math>

by the Følner sequence definition.