On the scope of the Effros theorem

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 $\mathsf{AC}$, there exists a non-Effros group; under $\mathsf{AD}$, every group is Effros; under $\mathsf{V=L}$, there exists a coanalytic non-Effros group. The above counterexamples will be graphs of discontinuous homomorphisms.

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