Definition of amenability by Følner sequences

amenabilitydefinitiongroup-theory

I am a little bit confused with the definition of amenability using the existence of Følner sequences. It seems that some people define a Følner sequence for certain classes of groups only. For example, Prof. Terence Tao in his notes writes:

Let $G = (G,\cdot)$ be a discrete, at most countable, group. A Følner sequence is a sequence $F_1, F_2, F_3, \ldots$ of finite subsets of $G$ with $\bigcup_{N=1}^\infty F_N = G$ with the property that $\lim_{N \to \infty} \frac{|g F_N \triangle F_N|}{|F_N|} = 0$ for all $g \in G$, where $\triangle$ denotes symmetric difference.

On the other hand, Prof. Cornelia Drutu gives a more straightforward definition, for any group $G$:

A sequence of non-empty subsets $F_n$ of $G$ is called a Følner sequence in $G$ if for every $g$ in $G$,
$$\lim\limits_{n \to \infty} \frac{|g F_n \triangle F_n|}{|F_n|}.$$

In both cases, the definition of amenability would be that $G$ (necessarily discrete and at most countable in the first case only) is amenable if it has at least one Følner sequence.

My questions are as follows.

  1. How are these two definitions related? Is there any reason to consider the topology on $G$ and limit oneself to countability? How are these assumptions relevant?
  2. If $G$ is countable and dicrete, are these two definitions equivalent?
  3. Is the condition $\bigcup_{N=1}^\infty F_N = G$ in the first definition necessary?
  4. Is the second definition equivalent to $G$ being non-paradoxical, under $\mathrm{AC}$? If so, is there a direct proof, without referring to other equivalent definitions of amenability?

Any help would be largely appreciated!

Best Answer

The condition that $\bigcup F_n=G$ is unusual, for instance $(\{1,\dots,n\})_{n\ge 0}$ is a reasonable Følner sequence in $\mathbf{Z}$.

If $(F_n)$ is Følner, however, one can modify it to get a covering sequence (i.e. $\bigcup F_n=G$). Just fix an enumeration $(g_n)$ of $G$ and set $F'_n=F_n\cup\{g_n\}$. It's still Følner.

Indeed if $G$ is finite, necessarily $F_n=G$ for large $n$. Otherwise, the added singleton makes a negligible contribution to the ratio.

For instance, $(\{-n\}\cup\{1,\dots,n\})_{n\ge 1}$ is Følner and covers $\mathbf{Z}$. This example illustrates that the covering condition is somewhat artificial.

Related Question