Citation: Burger, Marc, Narutaka Ozawa, and Andreas Thom. "On Ulam stability." arXiv preprint arXiv:1010.0565 (2010).
Web: https://arxiv.org/abs/1010.0565
Tags: Mathematical, Expository
In this paper, the authors give an overview of the the "Ulam stability problem". In this problem, one considers the notion of a delta-representation ("approximate representation"). These delta-representations are representations of your group on the space of unitary matrices on a Hilbert space which approximately satisfy the axioms of a representation. The goal is to find out when a delta-representation is approximately equal to an exact representation.
In general, not every approximate representation is approximately equal to an exact representation. A good example is free groups. The example is quite easy to construct in this case, and the authors do a good job explaining it.
The main positive theorem from this work is that amenable groups are (strongly) Ulam stable. This implies that quotients and subgroups of amenable groups will be Ulam stable. The proof that amenable groups are Ulam stable goes like this. First, you start with your approximate representation. Then, you take an average using the invariant mean on your amenable group, where this average is taken as pi(x)^dagger * pi(x * g) as x varies over elements of your group. It is a straightforward exercise to show that this averaged representation is quadratically closer to being an actual representation than your original one. It is not quite a unitary representation though, so you have to normalize it. At this point, you've now defined an abstract procedure for taking an approximate representation and making it more exact. Repeating this process ad infinitum gives the desired result.
Another point to be aware of from this work is "deformation rigidity". This is the problem of whether two exact representations could have small deformations from one to the other. An obvious example is conjugation - if you conjugate one representation by a unitary which is close to the identity then you will get another representation which is close to the identity. The solution to this is to consider the space of representations modulo conjugation. In this case it can be proved that the representations of amenable groups are all rigid - there are no small deformations to non-equivalent representations.