Logic Seminar

Several results regarding the topological and algebraic rigidity of maps and cocycles in the setting of Polish groups will be presented.
Firstly, suppose G is a Polish group acting continuously on a Polish space X, H is a Polish group and ψ:G×X→H is a cocycle that is continuous in the second variable. If ψ is either Baire measurable or is λ×μ-measurable with respect to Haar measure λ on G and a fully supported σ-finite Borel measure μ on X, then ψ is jointly continuous. Secondly, if π:G→S is a map from a locally compact Polish group G into an abstract semigroup S and such that there is a conull subset Ω⊆G×G satisfying π(gf)=π(g)⋅π(f) for all (g,f)∈Ω, then there is a homomorphism ψ:G→S agreeing with π almost everywhere. A similar statement holds for Baire category and further developments will be discussed.