Følner condition

Builds on Group

Definition

Let $G$ be a group, and for a subset $F \subseteq G$ and $g \in G$ let $gF = \set{gx : x \in F}$ . The group $G$ satisfies the Følner condition if the following holds. For every finite set $S \subseteq G$ and every $\varepsilon > 0$ , there is a finite nonempty set $F \subseteq G$ with $\abs{sF \,\triangle\, F} < \varepsilon \abs{F} \quad \text{for all } s \in S,$ where $\triangle$ is the symmetric difference. For a countable group, a Følner sequence is a sequence $(F_n)$ of finite nonempty subsets with $\abs{gF_n \triangle F_n} / \abs{F_n} \to 0$ for every $g \in G$ .

Remarks

Example: $G = \Z$ with $F_n = \set{0, 1, \ldots, n-1}$ . For $k \in \Z$ with $\abs{k} \le n$ , the sets $k + F_n$ and $F_n$ differ in $2\abs{k}$ points, so $\abs{(k + F_n) \triangle F_n} / n = 2\abs{k}/n \to 0$ . So $(F_n)$ is a Følner sequence.

Non-example: the free group $F_2$ on two generators fails the condition. Every finite nonempty $F \subseteq F_2$ has some generator $s$ with $\abs{sF \triangle F}$ at least a fixed positive fraction of $\abs{F}$ . This reflects that $F_2$ is not amenable.

Følner's theorem says that a discrete group satisfies the Følner condition exactly when it is amenable. For a locally compact group, one uses compact sets $F$ of positive Haar measure and compact sets $S$ instead.

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Group concept
Følner condition definition

Tag	`BP331JP8SF`
URI	`m2://BP331JP8SF`
Kind	definition
Writing	W0 stub
MSC	43A07, 20F65
Defines	Følner condition
Uses	Group
Changed	2 November 2026

Følner condition

Definition

Remarks

Reading path

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