Følner condition - Imported Knowlpedia Content

Definition

Let $G$ be a locally compact group with left Haar measure $\mu$ . A net $(F_i)$ of measurable, relatively compact subsets with $0<\mu(F_i)<\infty$ is a left Følner net if, for every compact subset $C\subseteq G$ ,

\sup_{g\in C}\frac{\mu(gF_i\mathbin{\triangle}F_i)} {\mu(F_i)}\longrightarrow0.

The group satisfies the Følner condition if it admits such a net. For a discrete group with counting measure, this says that finite nonempty subsets have boundary negligible relative to their size. A Følner sequence is a Følner net indexed by the positive integers.

Relation to amenability

The Følner condition gives almost invariant probability densities by normalizing the indicators $1_{F_i}$ , and hence produces a left-invariant mean through a weak-star cluster point. Conversely, standard Følner criteria recover nearly invariant sets from amenability under the usual locally compact hypotheses. This connects geometric boundary smallness with amenability Paterson, Chapters 1 and 4.

Examples and non-examples

For $\mathbb Z^d$ , the boxes $F_n=\{-n,\ldots,n\}^d$ form a Følner sequence because translating by a fixed finite set changes only a boundary layer of order $n^{d-1}$ , while $\lvert F_n\rvert$ has order $n^d$ . Expansion prevents the free group on two generators from having a Følner net; in particular, its balls have boundary comparable in size to their volume.

Conventions and scope

Some formulations use $\mu(CF_i\mathbin{\triangle}F_i)$ , others demand uniformity over $g\in C$ , and still others use the equivalent Leptin growth condition. Nets are the natural general form; the existence of a sequence can require countability assumptions. Right Følner conditions must account for the modular function when right translation does not preserve left Haar measure.

References

Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs 29, American Mathematical Society, 1988. AMS DOI record. Relevant: Følner conditions and their relationship with invariant means.
Erling Følner, “On groups with full Banach mean value,” Mathematica Scandinavica 3 (1955), 243–254. DOI record. Relevant: the original discrete-group condition.