On the scope of the Effros theorem

Abstract

All spaces (and groups) are assumed to be separable and metrizable. Jan van Mill showed that every analytic group G is Effros (that is, every continuous transitive action of G on a non-meager space is micro-transitive). We complete the picture by obtaining the following results: • Under AC, there exists a non-Effros group, • Under AD, every group is Effros, • Under V = L, there exists a coanalytic non-Effros group. The above counterexamples will be graphs of discontinuous homomorphisms.