This question is prompted by this one by Arturo Magidin: whether there exist varieties of groups in which the relatively free group of rank 2 is finite, and the relatively free group of rank 3 is infinite.

My question is: in which varieties of groups is it true that relatively free groups of bigger, countable or finite, rank embed into relatively free groups of smaller rank? This is so, for example, for the variety of all groups, and also for Burnside varieties (defined by the identity $x^n = 1$). On the other hand, this is not so for solvable or nilpotent varieties.

My knowledge on this is limited by the (old) book of H. Neumann "Varieties of Groups", and by paper of Shirvanyan about free Burnside groups. Probably more is known nowadays.

Of course, one can pose the same question also for other algebraic systems, for example, for algebras.

On coproducts in varieties, quasivarieties, and prevarieties, Algebra and Number Theory3(2009) 847-849, arXiv 0806.1750, MR 2011e:08014. $\endgroup$ – Arturo Magidin Oct 30 '11 at 23:29