I am currently studying the various characterizations of amenability, and I am in particular working with Følner sets: a group $G$ is amenable if and only if for every finite subset $S \subseteq G$ and every $\varepsilon >0$ there exists a non-empty finite subset $F \subseteq G$ such that $| sF \Delta F|\leq \varepsilon |F|$ for every $s \in S$.
I came across the concept of uniform amenability. The idea should be imposing that the size of an $\varepsilon$-Følner set for a generating set S can be uniformly bounded in terms of $\varepsilon$ and $|S|$. The definition is the following.
A group $G$ is uniformly amenable if there exists a function $f: \mathbb{R}_{> 0} \times \mathbb{N}$ such that, for every $\varepsilon >0$ and finite subsets $S \subseteq G$, there exists a finite subset $F \subseteq G$ such that:
- $|F| \leq f(\varepsilon, |S|)$;
- $|SF| \leq (1+ \varepsilon)|F|$.
From this, I deduce that condition $(2)$ is actually equivalent to the existence of Følner sets, so that, as written above, the uniform amenability is just amenability + a uniform bound on Følner sets.
However, I got stuck in trying to prove that the two conditions are equivalent. Any help in proving it will be highly appreciated. Thanks!
EDIT: I add a possible solution for the implication $(2) \Rightarrow$ amenability.
First of all, notice that if $S \subseteq T \subseteq G$ with $e \in T$, $S$ finite and $F \subseteq G$ finite, then $|SF \Delta F| \leq 2 |TF \Delta F|.$ Indeed, $$|SF \Delta F| = | SF \setminus F| + |F \setminus SF| = |SF \setminus F| + |F| - |S \cap SF| \leq $$ $$ \leq |SF \setminus F| + |SF| - |S \cap SF| = 2 |SF \setminus F| \leq 2 |TF \setminus F| = 2 |TF \Delta F|.$$ Let us fix $S \subseteq G$ finite and take $\varepsilon > 0$. Consider $T= S \cup \{e\}$: by $(2)$, there is a non-empty finite subset $F \subseteq G$ such that $|TF| \leq (1+\varepsilon)|F|.$ Let $s \in S$.
$$|sF \Delta F| \leq 2 |TF \Delta F| = 2|TF \setminus F| = 2 |TF| - 2|F| \leq 2(1+\varepsilon)|F| + 2|F| = 2\varepsilon|F|,$$
which implies amenability.
Any corrections on this reasoning and suggestions for the other implication are welcome.
EDIT 2: I also add a possible solution for the implication amenability $\Rightarrow (2)$.
Fix a finite subset $S \subseteq G$ and $\varepsilon > 0$. Apply the Følner condition for $\varepsilon'= \frac{\varepsilon}{|S|}$ and use the inclusion $SF \subseteq (\bigcup_{s \in S} sF \Delta F) \cup F.$ Then, $$|SF| \leq |(\bigcup_{s \in S} sF \Delta F) \cup F| \leq \sum_{s \in S} |sF \Delta F| + |F| \leq $$ $$\leq \sum_{s \in S} \frac{\varepsilon}{|S|} |F| + |F| = \varepsilon|F| + |F| = (1+ \varepsilon)|F|,$$ which implies $(2)$.