Some of the most important open problems in group theory concern whether all groups can be metrically approximated by certain classes of groups. In particular, it is currently unknown whether there exist groups that are not (linear) sofic, hyperlinear, or MF.
We introduce the general problem of approximation of groups, as well as the related problem of stability, which asks whether almost homomorphisms from a group are close to actual homomorphisms. We recall a cohomological tool for stability introduced by De-Chiffre, Glebsky, Lubotzky, and Thom, which was used to prove the existence of groups that are not Frobenius-norm approximated. Moreover, we propose a weaker notion of semi-stability, which relates the different classes of approximated groups, and discuss how this cohomological tool can be adapted for this purpose.
For more details, check out our website.
We look forward to seeing you in one or both sessions next Monday.
