A Cup Product Obstruction to Frobenius Stability

A countable discrete group $\Gamma$ 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 $\Gamma$ is finitely generated and a non-torsion element of $H^2(\Gamma;\mathbb{Z})$ can be written as a cup product of two elements in $H^1(\Gamma;\mathbb{Z})$ then $\Gamma$ 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 $H_3$. The argument is sufficiently general to show that the same condition implies non-stability in unnormalized Schatten $p$-norms for $1<p\le\infty$.