As mentioned before, by nil-2 groups we mean groups for which the commutators are central. Thus, for example, every element of the relatively free group on two letters x and y has a unique expression of the form:

   Since commutators are central, the commutator bracket [x,y] acts as a bilinear map from AxA to [G,G], where A is the abelianziation of G.

   This easily yields nontrivial dominions. With a bit more work and a theorem of Maier on amalgams of nil-2 groups, we obtain the following result:

   Theorem. Let G be a nil-2 group and H a subgroup of G. Let D be the subgroup of G generated by H and all elements of the form

where

Then D equals the dominion of H in G.

(See Dominions in varieties of nilpotent groups, Arturo Magidin, Comm. Algebra 28 no. 3 (2000), 1241-1270)

   This describes dominions for this variety, but there are a number of other interesting observations to make. Among them: (1) H is always normal in its dominion. (2) If G is finitely generated, then H is of finite index in its dominion. (3) D/H is a torsion abelian group.

   Relating our other question: For nonsurjective epimorphisms and saturated groups, we can easily verify from the description above that every group is saturated and thus no non-surjective epimorphisms exist in this variety (the latter can also be concluded from a general theorem of P.M. Neumann). As far as the strong and weak amalgamation conditions are concerned, Maier gave internal conditions for both; since they are a special case of my more general theorem for subvarieties of the class of all nil-2 groups, we direct the reader to that statement to see the conditions.

   In terms of amalgamation bases and absolute closures, the situation is a bit more complicated. D. Saracino described the weak and strong bases and proved that they are exactly the same groups. The description is a bit technical, but it can be adequately paraphrased by saying that a nil-2 group G is a strong amalgamation base if and only if it is a weak base if and only if the commutator subgroup of G equals the center and G has all the n-th roots it can possibly have modulo [G,G] (i.e. given an element x and n>0, either x has an n-th root in G/[G,G], or else it has no n-th root in any nil-2 overgroup of G). See Amalgamation bases for nil-2 groups, D. Saracino, Algebra Universalis 16 (1983), 47-62.

   Despite the fact that here strong and weak bases are the same, it turns out that there are more special bases than there are strong bases. The general description of the special bases is rather too complicated to state here (but see Absolutely closed nil-2 groups, Arturo Magidin, Algebra Universalis 42 no. 1/2 (1999) 61-77); but for example, although Saracino's descritpion implies that no nontrivial abelian group is a strong base, there are abelian groups which are special bases. In fact, an abelian group A is a special amalgamation base if and only if A/pA is trivial or cyclic for every prime number p; thus, for example, cyclic and divisible abelian groups are special amalgamation bases.

   With absolute closures the situation is slightly more disappointing. It turns out that the only nil-2 groups with an absolute closure are the absolutely closed groups (i.e. the special amalgamation bases). This result can be found at the end of my preprint Amalgams of nilpotent groups of class two.