A cup product obstruction to Frobenius stability

A countable discrete group Γ is said to be Frobenius stable if a function from the group that is "almost multiplicative" in the point Frobenius norm topology is "close" to a genuine unitary representation in the same topology. The purpose of this paper is to show that if Γ is finitely generated and a non-torsion element of H2(T; Z) can be written as a cup product of two elements in H1(T; Z), then Γ is not Frobenius stable. In general, 2-cohomology does not obstruct Frobenius stability. Some examples are discussed, including Thompson’s group F and Houghton’s group H3. The argument is sufficiently general to show that the same condition implies non-stability in unnormalized Schatten p-norms for 1 < p ≤ ∞.