Group graded algebras and almost polynomial growth

Let F be a field of characteristic 0, G a finite abelian group and A aG-graded algebra. We prove that A generates a variety of G-gradedalgebras of almost polynomial growth if and only if A has the samegraded identities as one of the following algebras:(1) FCp , the group algebra of a cyclic group of order p, where pis a prime number and p | |G|;(2) UTG2 (F ), the algebra of 2×2 upper triangular matrices over Fendowed with an elementary G-grading;(3) E, the infinite dimensional Grassmann algebra with trivial Ggrading;(4) in case 2 | |G|, EZ2 , the Grassmann algebra with canonical Z2-grading.