  This paper continues math.GR/0608302's study of amenability of affine
algebras (based on the notion of almost-invariant finite-dimensional subspace),
and applies it to graded algebras associated with finitely generated groups.
  Due to a mistake in Lemma 4.3, the main statements of the present paper
should not be considered as proven. The results in Sections 2 and 3 are
nevertheless valid.
