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Keller in "Amenable groups and varieties of groups" introduces uniformly amenable groups as groups such that there is a function $a: ]0,1[ \times \mathbb{N} \to \mathbb{N}$ such that for any finite subset $A \subset G$ and for any $k \in ]0,1[$ there is a finite subset $U$ of $G$ such that

  1. $|U|< a(k,|A|)$ and

  2. $\forall a \in A$, $|U \cap aU| > k |U|$.

Wysoczánski "On uniformly amenable groups" gives an example of a amenable group which is not uniformly amenable (it's $G = \oplus G_p$ where $G_p$ are the upper-triangular 3x3 matrices wit 1 on the diagonal and entries in $\mathbb{Z}/p\mathbb{Z}$).

Bożejko in "Uniformly amenable discrete groups" proves that groups of polynomial growth are uniformly amenable.

An example of a finitely generated [elementarily amenable] group which is not uniformly amenable is given in Lemma 3.24 of Ol'shanskii, Osin & Sapir's paper. This is a consequence of a result that [non-virtually cyclic] uniformly amenable groups satisfy a law by Corollary 6.17 of Druţu & Sapir

Another example is mentioned by de Cornulier and Mann (the linked paper contains a few questions on group laws): it is mentionned that the Grigorchuk group does not satisfy a law. In particular it may not be uniformly amenable.

Question: Are there examples of finitely generated amenable but not uniformly amenable groups which satisfy a law?

Another question (but it would imply a positive answer to a question from de Cornulier and Mann, so I assume it is open) is: is there a group of intermediate growth which is uniformly amenable?

Improve this question
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  • $\begingroup$ nilpotent $3\times 3$ matrices do not form a group $\endgroup$
    – YCor
    May 5, 2020 at 22:07
  • $\begingroup$ "non-uniformly amenable" means "not (uniformly amenable)" or "amenable but not uniformly amenable"? $\endgroup$
    – YCor
    May 5, 2020 at 22:09
  • $\begingroup$ sorry for the abusive language, I thought of the obvious nilpotent group in the 3x3 matrices and contracted it (into a nonsensical statement). Hopefully clarified both points... $\endgroup$
    – ARG
    May 5, 2020 at 22:17
  • $\begingroup$ But this group (the direct sum) seems to be uniformly amenable, since it is solvable, the author probably messed up something in his estimate. On the other hand such a direct sum but increasing both $p$ and the size of matrices, is non-uniformly-amenable. $\endgroup$
    – YCor
    May 6, 2020 at 6:13
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    $\begingroup$ The first f.g. elementary amenable groups satisfying no laws are due to BH Neumann in 1937 (subgroup of permutations of $\mathbf{Z}$ generated by infinite cycle $n\mapsto n+1$ and transposition $01$. That the ultrapower criterion implies that a group with no law is not uniformly amenable is immediate. So it should be attributed to Keller (1972). $\endgroup$
    – YCor
    May 6, 2020 at 6:17

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