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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.

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I assume you want to stick to countable or perhaps finite rank? –  Benjamin Steinberg Oct 30 '11 at 19:01
    
@Benjamin Steinberg: Yes. I will edit the question accordingly. –  Pasha Zusmanovich Oct 30 '11 at 19:59
    
This paper by George M. Bergman seems relevant: On coproducts in varieties, quasivarieties, and prevarieties, Algebra and Number Theory 3 (2009) 847-849, arXiv 0806.1750, MR 2011e:08014. –  Arturo Magidin Oct 30 '11 at 23:29
    
Here is a perhaps silly remark. Suppose that every countable group of the variety embeds in a 2-generated group of the variety, like happens in the variety of all groups. Then one can embed the free countably generated group in a 2-generated one. Writing the 2-generated group as a quotient of a relatively free group of rank 2 and splitting an appropriate restriction gives what you want. This leads to the question which varieties of groups have every countable member contained in a 2-generated member. –  Benjamin Steinberg Oct 31 '11 at 1:07
    
@Arturo Magidin: It is, in the context of universal algebra. Thanks! Adding "universal algebra" tag. –  Pasha Zusmanovich Oct 31 '11 at 7:43
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