Hyperlinearity versus flexible Hilbert Schmidt stability for property (T) groups

17.01.2023 15:00 - 17:00

Alon Dogon (Weizmann Institute)

 In these two talks, we will present and illustrate a phenomenon, commonly termed "stability vs. approximation", that has been present in several works in recent years. 

On the one hand, consider the following classical question: Given two almost commuting matrices/permutations, are they necessarily close to a pair of commuting matrices/permutations? This turns out to be a typical stability question for groups, which was introduced by G.N. Arzhantseva and L. Paunescu, and since then considered in different scenarios for general groups. 

On the other hand, the well known subject of approximation for groups is of central interest. Various metric approximation properties for groups have been defined by different mathematicians (including M. Gromov, A. Connes, F. Radulescu, E. Kirchberg....), resulting in notions such as sofic and hyperlinear groups, which have gained importance since their inception. Surprisingly, no counterexamples for failing soficity or hyperlinearity are known. A somewhat simple observation shows that a group that is both stable and approximable is residually finite. This yielded a successful strategy for constructing certain non-approximable groups by giving ones that are stable but not residually finite. 

In the introductory lecture we will discuss these notions precisely, and in the research part we will present classical residually finite groups, for which establishing (flexible Hilbert Schmidt) stability would still give non hyperlinear groups.

The same phenomenon is also shown to be generic for random groups in certain models.


G. Arzhantseva, Ch. Cashen