On the nonexistence of F{\o}lner sets

TL;DR

The paper demonstrates the nonexistence of F{46}lner sets in certain groups using model-theoretic methods, revealing limitations of first-order expressibility and properties of existentially closed groups.

Contribution

It constructs specific systems of equations showing no F{46}lner sets exist in some groups and connects this to model theory and properties of existentially closed groups.

Findings

No amenable group can be existentially closed.

Existentially closed groups cannot be exact, have the Haagerup property, or property (T).

F{46}lner set existence cannot be uniformly expressed in first-order logic for large enough parameters.

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.