On Ulam's Problem of Stability of Non-exact Homomorphisms

The paper is mainly devoted to Ulam's problem of stability of approximate homomorphisms, i.e., the problem of approximation of approximate group homomorphisms by exact homomorphisms. We consider the case when one of the groups is equipped with an invariant probability measure while the other one is a countable product of groups equipped with a (pseudo)metric induced by a submeasure on the index set. We demonstrate that, for submeasures satisfying a certain form of the Fubini theorem for the product with probability measures, the stability holds for all measurable homomorphisms. Special attention is given to the case of dyadic submeasures, or, equivalently, approximations modulo an ideal on the index set. Those ideals which give rise to Ulam–stable approximate homomorphisms have recently been distinguished as Radon–Nikodým (or RN) ideals. We prove that this class of ideals contains all Fatou ideals; in particular, all indecomposable ideals and Weiss ideals. Some counterexamples are also considered. In the last part of the paper we study the structure of Borel cohomologies of some groups; in particular, we show that the group $\mathrm H_{\mathrm {Bor}}^2(\mathbb R,G)$ is trivial for any at most countable group $G$.