Does there exist some sort of classification of subvarieties of $\mathfrak{N}_2$?

Question

Does there exist some sort of classification of subvarieties of $\mathfrak{N}_2$? Here $\mathfrak{N}_2$ stands for the variety of nilpotent groups of class $2$ (defined by the identity $[[x, y], z]$).

As we know that any variety is an intersection of a Burnside variety and a commutator variety (a variety in which all relatively free groups are torsion-free), it is sufficient to find all commutator subvarieties of $\mathfrak{N}_2$. Among them there are the variety of all abelian groups (defined by $[x, y]$ ). However, there definitely should be something else...

Answer 1

Yes, the full classification exists. It can be deduced from the classification of subvarieties of $\mathfrak{N}_3$ given in:

1. Bjarni Jónsson, Varieties of groups of nilpotency three, Notices Amer. Math. Soc. 13 (1966), 488.

2. V.N. Remeslennikov, Two remarks on 3-step nilpotent groups, Algebra i Logika (1965) no. 2, 59--65, MR 31:4838.

The first is an abstract, and has a typo (a $[y,x,z]$ should be $[y,x,x]$); the second contains the following classification:

Theorem. Every subvariety of $\mathfrak{N}_3$ corresponds to a $4$-tuple of nonnegative integers $(m,n,p,q)$ satisfying:

1. $n|m/\gcd(2,m)$,
2. $p|m$,
3. $q|p$,
4. $q|m/\gcd(6,m)$, and
5. $p|3q$;

corresponding to the identities $$x^m = [x,y]^n = [x,y,z]^p = [x,y,y]^q = [x,y,z,w] = 1.$$

From this, by taking $p=q=1$, we get:

Corollary. Every subvariety of $\mathfrak{N}_2$ may be defined by the identities $$x^m=[x,y]^n=[x,y,z]=1$$ for unique nonnegative integers $n$ and $m$ satisfying $n|m/\gcd(2,m)$.

Note: Every group of nilpotency class at most two is $2$-Engel, but not every $2$-Engel group has nilpotency class at most two; the $2$-Engel groups are not a subvariety of $\mathfrak{N}_2$; the $2$-Engel groups are the groups whose $2$-generated subgroups are in $\mathfrak{N}_2$, which contains $\mathfrak{N}_2$. (The original post, prior to editing, included an off-hand assertion that the variety of $2$-Engel groups would be realized as a subvariety of $\mathfrak{N}_2$, hence this comment...)