### 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 Π_{α}^{0}-complete, α≥3, and Σ_{α}^{0}-complete, α≥2, in each uncountable Polish group. Also, for every α<ω_{1} 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 language | English (US) |
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Pages (from-to) | 499-541 |

Number of pages | 43 |

Journal | Advances in Mathematics |

Volume | 199 |

Issue number | 2 |

DOIs | |

State | Published - Jan 30 2006 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Advances in Mathematics*,

*199*(2), 499-541. https://doi.org/10.1016/j.aim.2005.07.009