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 coset equivalence relation associated with G/H and 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 E1 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.