论文标题
有限生成的群体的拓扑零法律和基本等效性
A topological zero-one law and elementary equivalence of finitely generated groups
论文作者
论文摘要
令$ \ Mathcal G $表示有限生成的标记组的空间。 We give equivalent characterizations of closed subspaces $\mathcal S\subseteq \mathcal G$ satisfying the following zero-one law: for any sentence $σ$ in the infinitary logic $\mathcal L_{ω_1, ω}$, the set of all models of $σ$ in $\mathcal S$ is either meager or comeager.特别是，我们表明，零法律适用于与双曲线群及其概括相关的某些自然空间。作为一种应用，我们得到的是，无通用的无缘故双曲线基本上是等效的。对于没有非平凡有限的正常亚组的脱骨双曲线群，同样的主张也是。我们的论文具有大量的说明性组件。我们简化了与我们的工作相关的拓扑，逻辑和几何群体理论的一些已知结果和调查思想的证据。我们还讨论了一些开放问题。
Let $\mathcal G$ denote the space of finitely generated marked groups. We give equivalent characterizations of closed subspaces $\mathcal S\subseteq \mathcal G$ satisfying the following zero-one law: for any sentence $σ$ in the infinitary logic $\mathcal L_{ω_1, ω}$, the set of all models of $σ$ in $\mathcal S$ is either meager or comeager. In particular, we show that the zero-one law holds for certain natural spaces associated to hyperbolic groups and their generalizations. As an application, we obtain that generic torsion-free lacunary hyperbolic groups are elementarily equivalent; the same claim holds for lacunary hyperbolic groups without non-trivial finite normal subgroups. Our paper has a substantial expository component. We give streamlined proofs of some known results and survey ideas from topology, logic, and geometric group theory relevant to our work. We also discuss some open problems.
