Stability and Testability

We discuss various (still open) questions on approximations of finitely generated groups, focusing on finite-dimensional approximations such as residual finiteness and soficity. We survey our results on the existence, stability and quantification of...

We study stability problems with respect to families of groups equipped with bi-invariant ultrametrics, that is, metrics satisfying the strong triangle inequality. This property has very strong consequences, and this form of stability behaves very...

In [MIP*=RE by JNVWY] the authors construct a non-local game that resolves Tsirelson's problem to the negative and by that refute Connes' embedding conjecture (CEC). The game *-algebra (see e.g. [KPS]) enables one to construct a finitely presented *...

We argue that quantization, a mathematical model of the quantum classical correspondence, gives rise to approximate unitary representations of symplectomorphism groups. As an application, we get an obstruction to symplectic action of Lubotzky...

Constraint metric approximation is about constructing an approximation of a group G, when the approximation is already given for a subgroup H. Similarly, constraint stability is about lifting a representation of a group G, when the lift is already...

A discrete countable group is matricially stable if its finite dimensional approximate unitary representations are perturbable to genuine representations in the point-norm topology. We aim to explain in accessible terms why matricial stability for a...

This expository talk will try to bridge the first examples of "almost commuting" unitary matrices that are not almost "commuting unitaries" due to Voiculescu to a more sophisticated and very beautiful construction of examples by Gromov and Lawson in...

Some years ago, I proved with Shulman and Sørensen that precisely 12 of the 17 wallpaper groups are matricially stable in the operator norm. We did so as part of a general investigation of when group C∗C∗-algebras have the semiprojectivity and weak...

We show that the direct product of an infinite, finitely generated Kazhdan Property (T) group and a finitely presented, not residually finite amenable group admits no sofic approximation by expander graphs. Joint work with Andreas Thom.

A recent result of Becker, Lubotzky and Thom characterizes, for amenable groups, permutation stability in terms of co-soficity of invariant random subgroups (IRS). We will explain that for a class of amenable groups acting on rooted trees, including...