Borel Subgroups of Polish Groups

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 Πα-complete, α ≥ 3, and Σαcomplete, 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.