  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 $II_1$-factors we similarly define $II_1$-factor stability for
groups. Our main result is that all 1-relator groups with non-trivial center
are $II_{1}$-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
$\mathbb H_3$ 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.
