Finite groups and rings generating varieties with rapid growth

Let $A$ be a finite universal algebra. Then the orders of the $n$-generated free algebras $F_n$ in the variety (equational class) generated by $A$ satisfy G. Birkhoff’s inequality: $\mathrm{\mid }{F}_n\mathrm{\mid }\le \mathrm{\mid }A{\mathrm{\mid }}^{\mathrm{\mid }A{\mathrm{\mid }}^n}$ for $n=1,2,\dots$ It follows that ${\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}\sqrt[n]{\log \mathrm{\mid }{F}_n\mathrm{\mid }}\le \mathrm{\mid }A\mathrm{\mid }$. When $A$ is a finite group or a finite nonassociative algebra, we obtain a criterion for equality in this estimate; equivalently, a criterion for maximal growth of the sequence $\{\mathrm{\mid }{F}_n\mathrm{\mid }{\}}_{n=1}^{\mathrm{\infty }}$.