TY - JOUR

T1 - Borel subgroups of Polish groups

AU - Farah, Ilijas

AU - Solecki, Sławomir

N1 - Funding Information:
A part of the research for this paper was done while we were visiting the Fields Institute for Mathematical Sciences in Toronto in Fall 2002. We thank it for support during this time. We acknowledge the support received from NSERC (I.F.) and the National Science Foundation (USA) via Grants DMS-40313-00 01 (I.F.), and DMS-9803676 and DMS-0102254 (S.S.). We thank the referee for a very thorough reading of our paper and for a number of excellent comments.

PY - 2006/1/30

Y1 - 2006/1/30

N2 - 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.

AB - 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.

KW - Borel complexity of subgroups

KW - Densely divisible groups

KW - Maximal divisible subgroups

KW - Polish groups

KW - Polishable subgroups

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U2 - 10.1016/j.aim.2005.07.009

DO - 10.1016/j.aim.2005.07.009

M3 - Article

AN - SCOPUS:28944447244

SN - 0001-8708

VL - 199

SP - 499

EP - 541

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 2

ER -