On identities in the products of group varieties

Let ${\cal B}_n$ be the variety of groups satisfying the law $x^n=1$. It is proved that for every sufficiently large prime $p$, say $p>10^{10}$, the product ${\cal B}_p{\cal B}_p$ cannot be defined by a finite set of identities. This solves the problem formulated by C.K. Gupta and A.N. Krasilnikov in 2003. We also find the axiomatic and the basis ranks of the variety ${\cal B}_p{\cal B}_p$. For this goal, we improve the estimate for the basis rank of the product of group varieties obtained by G. Baumslag, B.H. Neumann, H. Neumann and P.M. Neumann long ago.