Averaging sequences and abelian rank in amenable groups (Q995357)

The abelian rank of a discrete group \(G\) is defined to be \[ r(G)= \sup\{n:G\text{ contains a subgroup isomorphic to }Z^n\}. \] An increasing Følner sequence \(\{F_n\}\) of finite subsets of \(G\) (i.e. \(\frac{1}{|F_n|}|gF_n\cap F_n|\to 1\) for all \(g\in G)\) is called a Tempel'man sequence if there is a constant \(C\) such that for every \(n\), \(|F_n^{-1}F_n|\leq C|F_n|\). In this paper, the author investigates the connection between the abelian rank of a countable amenable group and the existence of good averaging sequences. It is proved that if \(G\) is a group with finite abelian rank \(r(G)\), then \(2^{r(G)}\) is a lower bound on the constant associated to a Tempel'man sequence, and if \(G\) is abelian there is a Tempel'man sequence in \(G\) with this constant. If \(r(G)=\infty\), then \(G\) has no Tempel'man sequence.