Abstract:

Denote by $\mathcal{N}_3$  the variety of nilpotent groups of class at most $3$, by $\mathcal{M}$  the quasivariety generated by  the non-abelian $\mathcal{N}_3$-free  group $F$. Let $\mathcal{R}$ be an arbitrary  quasivariety such that $F\in\mathcal{R}$. Suppose that there  exits a  torsion free nilpotent group $G$ of class 3, $G\in \mathcal{R}\setminus \mathcal{M}$,  having the representation relative to $\mathcal{N}_3$ in which each defining relation  is a product of basic commutators  of weight 3 on three different variables or a commutator of the form $[x_i,x_j]$ (the last defining relations may be missing). It is proved that in this case the interval $[\mathcal{M},\mathcal{R}]$ in the lattice of quasivarieties of groups is continual.