Borel subgroups of Polish groups

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• 作者：Ilijas Farah and Sł ; awomir Solecki
• 关键词：Polish groups ; Polishable subgroups ; Densely divisible groups ; Maximal divisible subgroups ; Borel complexity of subgroups
• 刊名：Advances in Mathematics
• 出版年：2006
• 出版时间：30 January 2006
• 年：2006
• 卷：199
• 期：2
• 页码：499-541
• 全文大小：405 K

文摘

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 -complete, α3, and -complete, α2, 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.