A constructive proof that many groups with non-torsion $2$-cohomology are not matricially stable

Abstract

A discrete group is matricially stable if every function from the group to a complex unitary group that is “almost multiplicative” in the point-operator norm topology is “close” to a genuine unitary representation. It follows from a recent result due to Dadarlat that all amenable groups with non-torsion integral $2$-cohomology are not matricially stable, but the proof does not lead to explicit examples of asymptotic representations that are not perturbable to genuine representations. The purpose of this paper is to give an explicit formula, in terms of cohomological data, for asymptotic representations that are not perturbable to genuine representations for a class of groups that contains all finitely generated groups with a non-torsion $2$-cohomology class that corresponds to a central extension where the middle group is residually finite. This class includes polycyclic groups with non-torsion $2$-cohomology.