Stability of group relations under small Hilbert-Schmidt perturbations

If matrices almost satisfying a group relation are close to matrices exactly satisfying the relation, then we say that a group is matricially stable. Here “almost” and “close” are in terms of the Hilbert-Schmidt norm. Using tracial 2-norm on II1-factors we similarly define II1-factor stability for groups. Our main result is that all 1-relator groups with non-trivial center are II1-factor stable. Many of them are also matricially stable and RFD. For amenable groups we give a complete characterization of matricial stability in terms of the following approximation property for characters: each character must be a pointwise limit of traces of finite-dimensional representations. This allows us to prove matricial stability for the discrete Heisenberg group H3 and for all virtually abelian groups. For non-amenable groups the same approximation property is a necessary condition for being matricially stable. We study this approximation property and show that RF groups with character rigidity have it.