On varieties arising from the solution of the restricted Burnside problem (Q1612112)

Let \(h\) and \(n\) be positive integers, and let \(w\) be a multilinear commutator. Then let \({\mathfrak X}(w,n,h)\) be the class of groups \(G\) such that, first, the verbal subgroup \(w(G)\) has Hirsch-Plotkin height \(h\) (that is, \(w(G)\) has a normal series of length \(h\) with locally nilpotent factors) and, second, \(G\) satisfies the identity \(w^n=1\). It is proved that \({\mathfrak X}(w,n,h)\) is a variety. This is another generalization of the positive solution of the Restricted Burnside Problem. Earlier [Math. Proc. Camb. Philos. Soc. 132, No. 2, 193-196 (2002; see the preceding review Zbl 1007.20024)] the author proved this in the special case when \(w\) is a multilinear simple commutator. He also proposes the following interesting problems: let \(w\) be any group word and \(n\) a positive integer; is the class of groups \(G\) satisfying the identity \(w^n=1\) and having \(w(G)\) locally nilpotent (or locally finite) a variety? The proof in the present paper relies on \textit{E. I. Zel'manov}'s theorem [Lond. Math. Soc. Lect. Note Ser. 212, 567-585 (1995; Zbl 0860.20031)] stating that a finitely generated PI Lie algebra in which all commutators in the generators are ad-nilpotent is nilpotent. This powerful result is applied to the Lazard-Zassenhaus Lie algebra defined via dimension subgroups, which is a PI-algebra by a result of \textit{J. S. Wilson} and \textit{E. I. Zel'manov} [J. Pure Appl. Algebra 81, No. 1, 103-109 (1992; Zbl 0851.17007)].