On Ulam stability (Q1955863)

Let \(\pi:\Gamma\to U(H)\) be a map from a group \(\Gamma\) to the unitary group of a Hilbert space \(H\). It is called a \(\delta\)-quasi-representation if \(\|\pi(hg)-\pi(g)\pi(h)\|<\delta\) for any \(g,h\in\Gamma\). \(\Gamma\) is strongly Ulam stable (resp., Ulam stable) if, for sufficiently small \(\delta\), any of its quasi-representations (resp., finite-dimensional quasi-representations) is close to an actual representation. The authors establish (strong) Ulam stability or non-stability for certain classes of groups.