Question

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!

Answer 1

I learned a lot from reading Bestvina and Feighn's article Notes on Sela's work: Limit groups and Makanin-Razborov 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 2-complexes 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 Kharlampovich--Miasnikov 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.

Answer 2

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.