Averaging Sequences and Abelian Rank in Amenable Groups

We investigate the connection between the abelian rank of a countable amenable group and the existence of good averaging sequences (eg for the ergodic theorem). We show that if G is a group with finite abelian rank r(G), then 2 is a lower bound on the constant associated to a Tempel’man sequence, and if G is abelain there is a Tempel’man sequence in G with this constant. On the other hand, infinite rank precludes the existence of Tempel’man sequences and forces all tempered sequences to grow super-exponentially.