Operator algebra seminar
Speaker: Oren Becker, Einstein Institute of Mathematics
Title: Group theoretic stability
Abstract: Given two permutations A and B such that AB is close to BA, can we always find a commuting pair of permutations (A', B') such that A' is close to A and B' is close to B? Arzhantseva and Paunescu (2015) formalized this question and answered affirmatively. However, permutations almost satisfying A B^2=B^3 A are not always close to permutations which satisfy this equation exactly. We say that XY=YX is stable (in permutations), while X Y^2=Y^3 X is not. Our general question is which equations, or systems of simultaneous equations, are stable.
A productive way to study the stability of a given system of equations is to consider an equivalent problem regarding a corresponding finitely generated group. For example, the stability of XY=YX is equivalent to the following property of the group Z^2=<X,Y |XY=YX>: Every "almost action" of Z^2 on a finite set is close to a genuine action.
We will describe a relationship between stability, invariant random subgroups and sofic groups, giving, in particular, a characterization of stability among amenable groups. We will then show how to apply the above in concrete cases to prove and refute stability of some well-known classes of groups. Finally, we will discuss stability of groups with Kazhdan's property (T), some results on the quantitative aspect of stability and a connection to property testing in computer science.
Based on joint works with Alex Lubotzky, Andreas Thom and Jonathan Mosheiff.