Borel subgroups of Polish groups

Ilijas Farah, Sławomir Solecki

Research output: Contribution to journalArticlepeer-review


We study three classes of subgroups of Polish groups: Borel subgroups, Polishable subgroups, and maximal divisible subgroups. The membership of a subgroup in each of these classes allows one to assign to it a rank, that is, a countable ordinal, measuring in a natural way complexity of the subgroup. We prove theorems comparing these three ranks and construct subgroups with prescribed ranks. In particular, answering a question of Mauldin, we establish the existence of Borel subgroups which are Πα0-complete, α≥3, and Σα0-complete, α≥2, in each uncountable Polish group. Also, for every α<ω1 we construct an Abelian, locally compact, second countable group which is densely divisible and of Ulm length α + 1. All previously known such groups had Ulm length 0 or 1.

Original languageEnglish (US)
Pages (from-to)499-541
Number of pages43
JournalAdvances in Mathematics
Issue number2
StatePublished - Jan 30 2006


  • Borel complexity of subgroups
  • Densely divisible groups
  • Maximal divisible subgroups
  • Polish groups
  • Polishable subgroups

ASJC Scopus subject areas

  • General Mathematics


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