Varieties of group-graded algebras of proper central exponent greater than two

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, cG,δₙ( V), \, n ≥ 1. Here cG,δₙ( 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 expG,δ( V)=limn→∞√[n]cₙG,δ( 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 expG,δ( 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.