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, c(n)(G,delta)(V) , n >= 1 Here c(n)(G,delta)(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 [21] it was proved that exp(G,delta)(V) = lim(n ->infinity)n root c(n)(G,delta)(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,delta)(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 [11], we obtain a characterization of the varieties of proper central G-exponent equal to two. 2025 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).