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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Enumerated countable amenable groups admit Folner sequences

Statement

Let $G = {g_{1}, g_{2}, \dots}$ be an enumerated countable amenable group. Then $G$ admits a Folner sequence.

Facts & Assumptions

Given: An enumerated countable amenable group $G = {g_{1}, g_{2}, \dots}$ .

[L1]

A Folner sequence is a sequence of finite nonempty sets with vanishing relative symmetric-difference error for each fixed group element (Folner sequences for enumerated groups).

[L2]

Every amenable group satisfies the Folner condition (Under the ultrafilter lemma, the Folner condition is equivalent to amenability).

Proof

technique · direct

1.1

For each $n \geq 1$ , apply [L2] to the finite test set $S_{n} = {g_{1}, \dots, g_{n}}$ and the tolerance $1 / n$ . This gives a finite nonempty set $F_{n} \subseteq G$ with $∣ g_{i} F_{n} △ F_{n} ∣ < ∣ F_{n} ∣ / n$ for every $i \leq n$ .

L2givenchoose

2.1

Fix $i$ . Once $n \geq i$ , step 1.1 gives $∣ g_{i} F_{n} △ F_{n} ∣ / ∣ F_{n} ∣ < 1 / n$ , which tends to $0$ . Therefore $(F_{n})$ is a Folner sequence in the sense of [L1].

L1step 1.1∎

Depends on

Used by

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Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

C. Löh, Geometric Group Theory: An Introduction (2015 course version) (standard reference, not scraped)
Cornelia Drutu and Michael Kapovich, Lectures on Geometric Group Theory (standard reference, not scraped)