Given two permutations A and B which “almost” commute, are they “close” to permutations A’ and B’ which really commute? Arzhantseva and Paunescu (2015) formalized this question and answered affirmatively. This can be viewed as a property of the equation $XY=YX$, and turns out to be equivalent to the following property of the group $Z^2 = \langle X,Y \vert XY=YX \rangle$ Every “almost-everywhere locally-defined” action of $X^2$ is close to a genuine action of $Z^2$. This leads to the notion of locally testable groups (aka “stable groups”).

We will take the opportunity to say something about “local testability” in general, which is an important subject in computer science. We will then describe some results and methods to decide whether various groups are locally testable.
This will bring in some important notions in group theory, such as amenability, Kazhdan’s Property (T) and sofic groups.

Hiroki Matui defined the concept of the topological group of an étale groupoid. In the case where the groupoid comes from a one-sided shift of finite type, he proved that the associated topological full group is finitely generated. We will show how to re-interpret these groups as tree almost automorphism groups and thus obtain compactly generated groups almost acting on trees.