TITLE: need a good reference for introduction to elementary theory of groups
QUESTION [7 upvotes]: I've recently become interested in the elementary theory of groups due to Sela and Myasnikov-Kharlampovich's work with free groups.  I'd like a good introduction to the field of the elementary theory of groups, and in particular I'd like a reference to contain examples of group properties that cannot be read from a group's elementary theory.  For example, it seems that the statements "G is vritually abelian" or "G is hopfian" could not be expressed with first-order sentences, but I don't have enough knowledge yet to determine if such things are true.  Does anyone know such a reference?  Or, specifically, does someone know a proof of the fact that being hopfian (or some other group property) can't be read from a group's elementary theory?  Thanks!

REPLY [4 votes]: You might try Champetier and Guirardel's Limit groups as limits of free groups.
It has a short section (section 5) on elementary and universal theory, though perhaps none of the "non-examples" you're looking for.  It is, however, a pleasure to read and if you're interested in limit groups, Makanin-Razborov digrams, etc. I highly recommend it.