A Recipe for Almost-Representations of Groups that are Far from Genuine Representations

A group is said to be matricially (Frobenius) stable if every function from the group to unitary matrices that is "almost multiplicative" in the point operator (Frobenius) norm topology is "close" to a genuine unitary representation in the same topology. A result of Dadarlat shows that for a large class of groups, non-torsion even cohomology obstructs matricial stability. However, the proof doesn't generate explicit almost multiplicative maps that are far from genuine representations. In this paper, we compute explicit almost homomorphisms for all finitely generated groups with a non-torsion 2-cohomology class with a residually finite central extension. We use similar techniques to show that finitely generated nilpotent groups are Frobenius stable if and only if they are virtually cyclic, and that a finitely generated group with a non-torsion 2-cohomology class that can be written as a cup product of two 1-cohomology classes is not Frobenius stable.