The coset equivalence relation and topologies on subgroups

The paper studies the structure of the homogeneous space $G/H$, for $G$ a Polish group and $H< G$ a Borel, not necessarily closed subgroup of $G$, from the point of view of the theory of definable equivalence relations. It makes a connection between the complexity of the natural {\it coset equivalence relation} associated with $G/H$ and {\it Polishability} of $H$, that is, the possibility of introducing a Polish group topology on $H$ respecting its Borel structure. In particular, it is proved that if $H$ is an Abelian Borel subgroup of a Polish group $G$, then either $H$ is Polishable or ${\Bbb E}_1$ continuously embeds into the coset equivalence relation induced by $H$ on $G$. The same conclusion is shown to hold if $H$ is an increasing union of a sequence of Polishable subgroups of $G$.