Let F be a field of characteristic zero and let V be a variety of associative F-algebras graded by a finite abelian group G. To a variety V is associated a numerical sequence called the sequence of proper central G-codimensions, c^G, ₙ (V), \, n 1. Here c^G, ₙ (V) is the dimension of the space of multilinear proper central G-polynomials in n fixed variables of any algebra A generating the variety V. Such sequence gives information on the growth of the proper central G-polynomials of A and in LMR it was proved that exp^G, (V) =₍cₙ^G, (V) exists and is an integer called the proper central G-exponent. The aim of this paper is to characterize the varieties of associative G-graded algebras of proper central G-exponent greater than two. To this end we construct a finite list of G-graded algebras and we prove that exp^G, (V) >2 if and only if at least one of the algebras belongs to V. Matching this result with the characterization of the varieties of almost polynomial growth given in GLP, we obtain a characterization of the varieties of proper central G-exponent equal to two.
Benanti et al. (Mon,) studied this question.