ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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Under the ultrafilter lemma, the standard lamplighter group is amenable

Example

Assume the ultrafilter lemma. The lamplighter group

$L = (⨁_{n \in Z} Z / 2 Z) ⋊ Z$

is amenable.

Facts & Assumptions

Given: The semidirect product $L = (⨁_{Z} Z / 2 Z) ⋊ Z$ with the shift action of $Z$ on the lamp coordinates, and the ultrafilter lemma.

[L1]

An external semidirect product is the group built from an action by automorphisms ( The external semidirect product $N ⋊_{α} H$ ).

[L2]

Under the ultrafilter lemma, locally finite groups are amenable (Under the ultrafilter lemma, solvable groups and locally finite groups are amenable).

[L3]

Extensions of amenable groups are amenable (Extensions of amenable groups are amenable).

[L4]

Under the ultrafilter lemma, abelian groups are amenable (Under the ultrafilter lemma, abelian groups are amenable).

Verification

technique · direct

1.1

The base group $B = ⨁_{n \in Z} Z / 2 Z$ is locally finite, because a finitely generated subgroup is supported on finitely many coordinates and is therefore finite. Thus [L2] makes $B$ amenable.

L2givenalgebra

2.1

The quotient of $L$ by the normal base group $B$ is $Z$ , which is abelian and hence amenable by [L4]. Therefore [L3] applies to the semidirect product from [L1] and shows that the lamplighter group $L$ is amenable.

L1L3L4step 1.1∎

Depends on

Used by

Under the ultrafilter lemma, amenability does not imply subexponential growth Counterexample

Dependency tree · two levels

15 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)