论文标题
自由作用在线作用的组的抗分类结果
Anti-classification results for groups acting freely on the line
论文作者
论文摘要
我们从描述性集理论的角度探索了可数的阿基米德群体。我们为给定的可计数组$ g $介绍了阿基米德左订购$ \ mathrm {ar}（g）$ $可分类。然后，我们分析了可计数有序的阿基米德群体的同构关系，并根据Hjorth，Kechris和Louveau的层次结构固定其复杂性。特别是，我们表明其潜在类不是$ \boldsymbolπ^0_3 $。这种拓扑约束阻止使用可计数的真实子集对阿基米德组进行分类。我们获得了双重性关系的类似结果，并且我们考虑了循环有序组的类似问题，以及O最低结构（例如有序的可划分的Abelian群体）和实际的封闭场。我们的证据结合了阿基米德群体，鲍尔等效关系理论的经典结果，并在基本的科恩模型中分析了可定义的集合以及Zermelo-fraenkel设置的其他模型，而无需选择。
We explore countable ordered Archimedean groups from the point of view of descriptive set theory. We introduce the space of Archimedean left-orderings $\mathrm{Ar}(G)$ for a given countable group $G$, and prove that the equivalence relation induced by the natural action of $\mathrm{GL}_2(\mathbb{Q})$ on $\mathrm{Ar}(\mathbb{Q}^2)$ is not concretely classifiable. Then we analyze the isomorphism relation for countable ordered Archimedean groups, and pin its complexity in terms of the hierarchy of Hjorth, Kechris and Louveau. In particular, we show that its potential class is not $\boldsymbolΠ^0_3$. This topological constraint prevents classifying Archimedean groups using countable subsets of reals. We obtain analogous results for the bi-embeddability relation, and we consider similar problems for circularly ordered groups, and o-minimal structures such as ordered divisible Abelian groups, and real closed fields. Our proofs combine classical results on Archimedean groups, the theory of Borel equivalence relations, and analyzing definable sets in the basic Cohen model and other models of Zermelo-Fraenkel set theory without choice.
