|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||Abstract:||
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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