I've recently become interested in the elementary theory of groups due to Sela and MyasnikovKharlampovich'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 firstorder 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!
2 Answers
I learned a lot from reading Bestvina and Feighn's article Notes on Sela's work: Limit groups and MakaninRazborov diagrams. It's not a broad introduction to elementary theory, but it does express some of Sela's ideas quite succinctly. You may need a background in geometric group theory (specifically, in understanding laminations on 2complexes or group actions on $\mathbb{R}$trees) to get the most out of it.
Regarding the question of whether or not the Hopf property is elementary: the answer is obviously `no' if you allow infinitely generated examples. Indeed, consider the free group on countably many generators, $F_\infty$. The elementary theory is completely determined by the set of finitely generated subgroups, so $F_\infty$ is elementary equivalent to $F_2$. But $F_2$ is Hopfian and $F_\infty$ is not.
EDIT: I'm getting a little nervous about the claim that the elementary theory is determined by the list of finitely generated subgroups. However, Sela and KharlampovichMiasnikov proved that the natural inclusions $F_n\subseteq F_{n+1}$ are elementary embeddings, from which it does indeed follow that $F_\infty$ is elementarily equivalent to $F_2$.
I don't know a finitely generated example, although I agree that one must surely exist.

3$\begingroup$ Henry is too bashful to point it out, but he wrote a very nice set of solutions to the exercises in BestvinaFeighn's paper (some of which are pretty hard, so they make BF's paper much easier to read). They are located here : arxiv.org/abs/math.GR/0604137 $\endgroup$ Feb 14, 2011 at 21:42


$\begingroup$ Thanks for the reference Henry. To get a general introduction to the elementary theory of groups I have looked at some Model Theory books, but these books rarely reference groups directly. Is it possible that people haven't thought that much about what group properties can be read from the elementary theory of a group? $\endgroup$– danFeb 15, 2011 at 3:34

1$\begingroup$ dan, people have certainly thought about these things. My impression is that to construct finitely generated counterexamples to show that a certain property isn't elementarily is just, in general, a very difficult problem. Logicians seem to like to construct elementarily equivalent groups via ultrapowers, which of course give infinitely generated examples in general. From a logical point of view, finitely generated examples aren't particularly natural, since being finitely generated is not an elementary property! $\endgroup$– HJRWFeb 15, 2011 at 18:25
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 "nonexamples" you're looking for. It is, however, a pleasure to read and if you're interested in limit groups, MakaninRazborov digrams, etc. I highly recommend it.

$\begingroup$ Thanks for the reference Jeremy. That paper is what got me interested in the elementary theory stuff, but their introduction to the theory is very brief. $\endgroup$– danFeb 15, 2011 at 3:30