Bounded characteristic classes and flat bundles

Abstract

Let $G$ be a connected Lie group. We show that all characteristic classes of $G$ are bounded—when viewed in the cohomology of the classifying space of the group $G$ with the discrete topology—if and only if the derived group of the radical of $G$ is simply connected in its Lie group topology. We also give equivalent conditions in terms of stable commutator length and distortion.