Følner sequence

Examples

• Any finite group ${\displaystyle G}$ trivially has a Følner sequence ${\displaystyle F_i=G}$ for each ${\displaystyle i}$.
• Consider the group of integers, acting on itself by addition. Let ${\displaystyle F_i}$ consist of the integers between ${\displaystyle -i}$ and ${\displaystyle i}$. Then ${\displaystyle g{F}_i}$ consists of integers between ${\displaystyle g-i}$ and ${\displaystyle g+i}$. For large ${\displaystyle i}$, the symmetric difference has size ${\displaystyle 2g}$, while ${\displaystyle F_i}$ has size ${\displaystyle 2i+1}$. The resulting ratio is ${\displaystyle 2g/(2i+1)}$, which goes to 0 as ${\displaystyle i}$ gets large.
• With the original definition of Følner sequence, a group has a Følner sequence if and only if it is countable and amenable.
• A locally compact group has a Følner sequence (with the generalized definition) if and only if it is amenable and second countable.

Proof of amenability [citation needed]

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

${\displaystyle \mu (A)=\underset{i\to \mathrm{\infty }}{lim}\frac{|A\cap F_i|}{|F_i|}.}$

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

${\displaystyle \mu (A)=U-lim\frac{|A\cap F_i|}{|F_i|}.}$

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

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

   ${\displaystyle \left|\frac{|gA\cap F_i|}{|F_i|}-\frac{|A\cap F_i|}{|F_i|}\right|=\left|\frac{|A\cap g^{-1}{F}_i|}{|F_i|}-\frac{|A\cap F_i|}{|F_i|}\right|}$

   ${\displaystyle \le \frac{|A\cap (g^{-1}{F}_i\mathrm{△}{F}_i)|}{|F_i|}\to 0}$

by the Følner sequence definition.