Averaging sequences and abelian rank in amenable groups

Abstract

We investigate the connection between the abelian rank of a countable amenable group and the existence of good averaging sequences (e.g. for the pointwise ergodic theorem). We show that if $G$ is a group of abelian rank $r(G)$ then any Tempel'man sequence must have constant at least $2^{r(G)}$ and if $G$ is abelian this constant is achieved. On the other hand, infinite rank excludes the existence of Tempel'man sequences and forces all tempered sequences to grow super-exponentially.