On the scope of the Effros theorem

TL;DR

This paper explores the conditions under which groups are Effros, showing that set-theoretic assumptions like AC, AD, and V=L influence whether all groups are Effros or if counterexamples exist, often involving discontinuous homomorphisms.

Contribution

It completes the characterization of Effros groups by demonstrating the existence or non-existence of non-Effros groups under various set-theoretic axioms.

Findings

Under AC, non-Effros groups exist.

Under AD, all groups are Effros.

Under V=L, non-Effros groups exist as coanalytic sets.

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 $\mathsf{AC}$, 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.