Семинар "Kourovka Forum"

In group theory, interesting statements about a group usually can't be expressed in the language of first-order logic. It turns out, however, that some groups can actually be determined by their first-order properties, or, even more strongly, by a single first-order sentence. In the latter case the group is said to be finitely axiomatizable.

I will describe some examples of this phenomenon (joint work with A. Nies and K. Tent). One family of results concerns axiomatizability of $p$-adic analytic pro-$p$ groups, within the class of all rofinite groups. Another main result is that for an adjoint simple Chevalley group of rank at least 2 and an integral domain $R$; the group $G(R)$ is bi-interpretable with the ring $R$. This means in particular that first-order properties of the group $G(R)$ correspond to first-order properties of the ring $R$. As many rings are known to be finitely axiomatizable we obtain the corresponding result for many groups; this holds in particular for every finitely generated group of the form $G(R)$.

Let $A$ and $B$ be two permutations in $Sym(n)$ which "almost commute" - are they a small deformation of permutations that truly commute?

More generally, if $R$ is a system of word-equations in variables $X =(x_1, \cdots ,x_d)$ and $A =(A_1, \cdots , A_d)$ permutations which are nearly solution; are they near true solutions?

It turns out that the answer to this question depends only on the group presented by the generators $X$ and relations $R$.

This leads to the notions of "stable groups" and "testable groups". We will present a few results and methods which were developed in recent years to check whether a group is stable\testable. We will also describe the connection of this subject with property testing in computer science, with the long-standing problem of whether every group is sofic and with IRS's ( =invariant random subgroups). A number of open questions will be presented.