Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Under the ultrafilter lemma, amenability does not imply subexponential growth
Statement refuted
Every amenable finitely generated group has subexponential growth.
Facts & Assumptions
Given: The false claim above and the ultrafilter lemma.
Exponential growth is one of the growth types in the standard comparison hierarchy (Polynomial, subexponential, exponential, and intermediate growth).
Under the ultrafilter lemma, the standard lamplighter group is amenable (Under the ultrafilter lemma, the standard lamplighter group is amenable).
Counterexample
Let with generators for the shift and for toggling the lamp at the origin. For every subset , the element obtained by walking from to , toggling exactly the lamps in on the way, has word length at most ; different subsets give different group elements.
Therefore the ball of radius contains at least elements, so has exponential growth in the sense of [L1]. Together with [L2], this amenable group refutes the statement.
Depends on
Used by
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