The situation regarding dominions in solvable groups is not much better than that concerning amalgams in general, and is certainly much below the situation for nilpotent groups. We do have some interesting results, particularly for the metabelian groups (groups which are extensions of abelian or abelian; equivalently, groups for which the commutator subgroup is abelian; also equivalently, solvable of length at most two).

   Let G be the free metabelian group on two generators x and y, and let H be the subgroup generated by the commutators [x,y], [x,y,x] and [x,y,y]. The commutator subgroup of G contains a free abelian group if infinite rank, and H is a finitely generated subgroup of it.

   Well, perhaps surprisingly, it turns out that H dominates all of [G,G]; that is, the dominion of H in G equals the commutator subgroup of G. Part of the reason this may seem surprising is that this means that H, which is a direct summand of rank three of that countably generated free abelian group actually dominates that entire abelian subgroup and more. And yet, it is normal in its dominion. Somehow, the structure imposed on [G,G] by the actions of x and y is so rigid that its image is completely determined by the comparatively small subgroup H.

   A similar argument works if we consider the variety of all extensions of abelian groups of exponent n by abelian groups of exponent m (a subvariety of the variety of all metabelian groups).

   Another reason for my surprise at these results comes to us from Universal Algebra. If we consider two varieties V and W, we can form their product VW which is the variety of all groups which are extensions of a V-group by a W-group. From a famous theorem of B.H. Neumann, Hanna Neumann, Peter Neumann, and G. Baumslag, every variety of groups can be written uniquely as a finite product of indecomposable varieties (those that cannot be written as a product involving varieties other than the trivial and the total variety), in a unique way, in a unique order. The variety of metabelian groups, for example, is the product AA of two copies of the variety of all abelian groups A.

   I thought at first that dominions in a product variety might be completely determined by dominions in the factors. (For more on this, see the section on more general classes). However, if this were the case then, since dominions in abelian groups are trivial (and in a very strong way: every subgroup is normal, so it will remain dominion-closed in any variety containing G), it would have been expected that dominions in metabelian groups (and in fact, in any variety of solvable groups) would necessarily be trivial. But this is very much not the case, as seen above! In fact, a comparatively small group may have a rather large dominion. So the product variety somehow introduces a rather strong rigidity into the groups that lie there.

   Unfortunately, I still do not have a complete description of dominions in this variety (although I have several sufficient conditions for an element to be in the dominion; see Dominions in varieties of nilpotent groups, A. Magidin, Comm. Algebra 28 no. 3 (2000), 1241-1270). I am trying to use Leinen's theorem to obtain such a description, but have so far been unsuccessful.