Семинары ИМ СО РАН

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.