gr.group theory - Why are abelian groups amenable?

Here is a simpler argument, combining 1--6 into one step.

Let $G$ be a countable abelian group generated by $x_1,x_2,ldots$. Then a Følner sequence is given by taking $S_n$ to be the pyramid consisting of elements which can be written as

$a_1x_2+a_2x_2+cdots+a_nx_n$ with $lvert a_1rvertleq n,lvert a_2rvertleq n-1,ldots,lvert a_nrvertleq 1$.

The invariant probability measure is then defined by $mu(A)=underset{omega}{lim}lvert Acap S_nrvert / lvert S_nrvert$ as usual.

A more natural way to phrase this argument is:

1. The countable group $mathbb{Z}^infty$ is amenable.


2. All countable abelian groups are amenable, because amenability descends to quotients.

But I would like to emphasize that there is really only one step here, because the proof for $mathbb{Z}^infty$ automatically applies to any countable abelian group. This two-step approach is easier to remember, though. (The ideas here are the same as in my other answer, but I think this formulation is much cleaner.)

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2016 Edit: Here is an argument to see that $S_n$ is a Følner sequence. It is quite pleasant to think about precisely where commutativity comes into play.

Fix $gin G$ and any finite subset $Ssubset G$. We first analyze the size of the symmetric difference $gSbigtriangleup S$. Consider the equivalence relation on $S$ generated by the relation $xsim y$ if $y=x+g$ (which is itself neither symmetric, reflexive, or transitive). We will call an equivalence class under this relation a "$g$-string". Every $g$-string consists of elements $x_1,ldots,x_kin S$ with $x_{j+1}=x_j+g$.

The first key observation is that $lvert gSbigtriangleup Srvert$ is at most twice the number of $g$-strings. Indeed, if $zin S$ belongs to $gSbigtriangleup S$, then $z$ must be the "leftmost endpoint" of a $g$-string; if $znotin S$ belongs to $gSbigtriangleup S$, then $z-g$ must be the "rightmost endpoint" of a $g$-string; and each $g$-string has at most 2 such endpoints (it could have 1 if the endpoints coincide, or 0 if $g$ has finite order).

Our goal is to prove for all $gin G$ that $frac{lvert gS_n bigtriangleup S_nrvert}{lvert S_nrvert}to 0$ as $nto infty$. Since $lvert abSbigtriangleup Srvertleqlvert abSbigtriangleup bSrvert+lvert bSbigtriangleup Srvert= lvert aSbigtriangleup Srvert+lvert bSbigtriangleup Srvert$, it suffices to prove this for all $g_i$ in a generating set.

By the observation above, to prove that $frac{lvert g_iS_n bigtriangleup S_nrvert}{lvert S_nrvert}to 0$, it suffices to prove that $frac{text{# of $g_i$-strings in $S_n$}}{lvert S_nrvert}to 0$. Equivalently, we must prove that the reciprocal $frac{lvert S_nrvert}{#text{ of $g_i$-strings in $S_n$}}$ diverges, or in other words that the average size of a $g_i$-string in $S_n$ diverges.

We now use the specific form of our sets $S_n={a_1g_1+cdots+a_ng_n,|, lvert a_irvertleq n-i}$. For any $i$ and any $n$, set $k=n-i$ (so that $lvert a_irvertleq k$ in $S_n$). The second key observation is that every $g_i$-string in $S_n$ has cardinality at least $2k+1$ unless $g_i$ has finite order. Indeed given $xin S_n$, write it as $x=a_1g_1+cdots+a_ig_i+cdots+a_ng_n$; then the elements $a_1g_1+cdots+bg_i+cdots+a_ng_nin S_n$ for $b=-k,ldots,-1,0,1,ldots,k$ belong to a single $g_i$-string containing $x$. If $g_i$ does not have finite order, these $2k+1$ elements must be distinct. This shows that the minimum size of a $g_i$-string in $S_n$ is $2n-2i+1$, so for fixed $g_i$ the average size diverges as $nto infty$.

When $g_i$ has finite order $N$ this argument does not work (a $g_i$-string has maximum size $N$, so the average size cannot diverge). However once $N<2k+1$, the subset containing the $2k+1$ elements above is closed under multiplication by $g_i$. In other words, once $ngeq i+N/2$ the set $S_n$ is $g_i$-invariant, so $lvert g_iS_nbigtriangleup S_nrvert=0$.

I'm grateful to David Ullrich for pointing out that this claim is not obvious, since the quotient of a Følner sequence need not be a Følner sequence (Yves Cornulier gives an example here).