论文标题
枚举组空间中的通用代数特性
Generic algebraic properties in spaces of enumerated groups
论文作者
论文摘要
我们介绍和研究波兰拓扑上的波兰拓扑，在可计数的各个空间上，枚举的群体只是一个基础集的组是一组自然数。使用基本工具和联合群体理论的众所周知的示例，结合了Baire类别定理，我们获得了许多结果，表明群体理论中的几种现象是通用的。实际上，我们为分析群体理论中各种众所周知的问题提供了一个新的拓扑框架。我们还提供了这些空间中的通用性之间的联系，即有限生成的群体和模型理论强迫的一词问题。使用这些连接，我们研究了一个自然的问题：枚举群体的某个空间何时包含一致的同构类别？我们获得了足够的条件，使我们能够对所有枚举组的空间和左有序枚举组的空间的否定问题回答这个问题。我们记录了与这些注意事项有关的几个开放问题。
We introduce and study Polish topologies on various spaces of countable enumerated groups, where an enumerated group is simply a group whose underlying set is the set of natural numbers. Using elementary tools and well known examples from combinatorial group theory, combined with the Baire category theorem, we obtain a plethora of results demonstrating that several phenomena in group theory are generic. In effect, we provide a new topological framework for the analysis of various well known problems in group theory. We also provide a connection between genericity in these spaces, the word problem for finitely generated groups and model-theoretic forcing. Using these connections, we investigate the natural question: when does a certain space of enumerated groups contain a comeager isomorphism class? We obtain a sufficient condition that allows us to answer the question in the negative for the space of all enumerated groups and the space of left orderable enumerated groups. We document several open questions in connection with these considerations.
