Varieties with at most quadratic growth

Let V be a variety of non-necessarily associative algebras over a field of characteristic zero. The growth of V is determined by the asymptotic behavior of the sequence of codimensions cn(V), n = 1, 2, …, and here we study varieties of polynomial growth. Recently in [16], for any real number α, 3 < α < 4, a variety V was constructed satisfying C1nα < cn(V) < C2nα, for some constants C1, C2. Motivated by this result here we try to classify all possible growth of varieties V such that cn(V) < Cnα, with 0 < α < 2, for some constant C. We prove that if 0 < α < 1 then, for n large, cn(V) ≤ 1, whereas if V is a commutative variety and 1 < α < 2, then limn→∞ logncn(V) = 1 or cn(V) ≤ 1 for n large enough.