Sublattices of verbal subgroups. (Q2820742)

The authors consider four classes of verbal subgroups of a free group \(F\) of rank 2: \(\{VN\mathrm{-verbal}\}\subseteq\{P\mathrm{-verbal}\}\subseteq\{R\mathrm{-verbal}\}\subseteq\{M\mathrm{-verbal}\}\). Four types of varieties \(\mathrm{var}(F/V)\) arise: \(VN\)-varieties have their 2-generator groups virtually nilpotent, \(P\)-varieties satisfy positive laws, every 2-generator group in an \(R\)-variety has a finitely generated commutator subgroup, \(M\)-varieties have no subvariety which is the product of the variety of all abelian groups of exponent \(p\) by the variety of all abelian groups for any prime \(p\). It is shown that each of these classes of verbal subgroups forms a sublattice of the lattice of subgroups in \(F\). Three questions are posed.