Authors: Farah, Ilijas 
Solecki, Sławomir
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Borel subgroups of Polish groups
Journal: Advances in Mathematics
Volume: 199
Issue: 2
First page: 499
Last page: 541
Issue Date: 30-Jan-2006
Rank: M21a
ISSN: 0001-8708
DOI: 10.1016/j.aim.2005.07.009
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.
Keywords: Borel complexity of subgroups | Densely divisible groups | Maximal divisible subgroups | Polish groups | Polishable subgroups
Publisher: Elsevier
Project: National Science Foundation (USA), Grants DMS-40313-00 01, DMS-9803676 and DMS-0102254

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