Stability of asymptotic representations, cohomology vanishing, and non-approximable groups

Abstract: Since the very beginning of the study of groups, groups were studied by looking at their orthogonal and unitary representations. It is very natural to relax the notion of a representation and require the group multiplication to be preserved only up to little mistakes in a suitable metric. I will provide (based on joint work with Marcus De Chiffre, Alex Lubotzky and Lev Glebsky) the first examples of countable groups which are not Frobenius-approximated, i.e. does not admit sufficiently many almost representations to separate the group elements. Our strategy is to use higher-dimensional cohomology vanishing phenomena to prove that any Frobenius-almost homomorphism into finite-dimensional unitary groups is close to an actual homomorphism and combine this with existence results of certain non-residually finite central extensions of lattices of higher rank. We ultimately rely on work of Garland, Ballmann-Swiatkowski, Deligne, Rapinchuk and others. Everything will be motivated and definition will be explained in detail.