On the nonexistence of Følner sets

Abstract: We show that there is $n\in \mathbf N$, a finite system $\Sigma(\vec x,\vec y)$ of equations and inequations having a solution in some group, where $\vec x$ has length $n$, and $\epsilon>0$ such that: for any group $G$ and any $\vec a\in G^n$, if the system $\Sigma(\vec a,\vec y)$ has a solution in $G$, then there is no $(\vec a,\epsilon)$-F{\o} lner set in $G$. The proof uses ideas from model-theoretic forcing together with the observation that no amenable group can be existentially closed. Along the way, we also observe that no existentially closed group can be exact, have the Haagerup property, or have property (T). Finally, we show that, for $n$ large enough and for $\epsilon$ small enough, the existence of $(F,\epsilon)$-F{\o} lner sets, where $F$ has size at most $n$, cannot be expressed in a first-order way uniformly in all groups.