Extensions of amenable groups

Question

Let $1 \to N \to G \to Q \to 1$ be an extension of (discrete) groups, where $N$ and $Q$ are amenable. Using the fixed-point theorem, I know how to show that $G$ is amenable. However, I was wondering if there is a proof which only uses the Følner property, namely $G$ is amenable if and only if: for every finite $S \subset G$ and for every $\epsilon > 0$ there exists some finite $A \subset G$ such that $|SA \Delta A| < \epsilon |A|$. I am asking this since in some texts this is used as the actual definition of an amenable group.

Answer 1

Here's a direct proof.

Let $G$ be a group with $N$ amenable normal subgroup, such that $G/N$ is amenable; write $p:G\to G/N$. Let $S$ be a finite subset of $G$ and $\varepsilon>0$.

There exists a nonempty finite subset $F$ of $G$ such that $p(F)\equiv_\varepsilon p(sF)$ for all $s\in S$, and $p|_F$ is injective. Here $A\equiv_t B$, for finite nonempty subsets $A,B$ means $|A\Delta B|\le t|A|$ (beware that this is a symmetric relation only when $|A|=|B|$).

For $s\in S$ and $g\in F$ such that $p(sg)\in p(F)$, write $sg=f(s,g)\nu(s,g)$, with $\nu(s,g)\in N$ and $f(s,g)\in F$. Let $T\subset N$ be the (finite) set of $\nu(s,g)$ when $(s,g)$ range over such pairs

Let $M$ be a nonempty finite subset of $N$ such that $M\equiv_\varepsilon tM$ for all $t\in T$.

Then, for $s\in S$ and $g\in F$ such that $p(sg)\in p(F)$, we have $sgM=f(s,g)\nu(s,g)M\equiv_\varepsilon f(s,g)M$. Hence, if $F_s\subset F$ is the union of such $g$ and $F'_s\subset F$ is the union of such $f(s,g)$, we have $sF_sM\equiv_\varepsilon F'_sM$.

Also, we have $sFM\equiv_\varepsilon sF_sM$ and $FM\equiv_\varepsilon F'_sM$. So $$|sFM\smallsetminus FM|\le |sFM\smallsetminus F'_sM|\le |sFM\smallsetminus sF_sM|+|sF_sM\smallsetminus F'_sM|\le \varepsilon|FM|+\varepsilon|F_sM|\le 2\varepsilon|FM|$$ and similarly $|FM\smallsetminus sFM|\le 2\varepsilon|FM|$. Therefore $|FM\Delta sFM|\le 4\varepsilon$.

So there's a $(S,4\varepsilon)$-invariant subset. Hence $G$ is amenable.