The coset equivalence relation and topologies on subgroups

Slawomir Solecki

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish (US)
Pages (from-to)571-605
Number of pages35
JournalAmerican Journal of Mathematics
Volume131
Issue number3
DOIs
StatePublished - 2009

ASJC Scopus subject areas

  • Mathematics(all)

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