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 ${\mathcal{L}}_{{\omega }_1,\omega }$, the set of all models of σ in $\mathcal{S}$ is either meager or comeager. In particular, we prove that the zero-one law holds for certain natural spaces associated to hyperbolic groups and their generalizations. As an application, we show 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.

让 $\mathcal{G}$表示有限生成的标记组的空间。我们给出封闭子空间的等效特征$\mathcal{小号}\subseteq \mathcal{G}$满足以下零一定律：对于不定式逻辑中的任何句子σ${\mathcal{大号}}_{{\omega }_{\mathrm{1个}}，\omega }$，是σ中所有模型的集合$\mathcal{小号}$是微不足道的还是虚弱的。特别地，我们证明零一定律适用于某些与双曲群及其推广有关的自然空间。作为一个应用，我们证明了无扭转泛型双曲线基团在本质上是等效的。对于没有非平凡有限正态子群的腔双曲群，该主张也成立。我们的论文有很多说明性的内容。我们提供一些已知结果的简化证据，并从与我们的工作相关的拓扑，逻辑和几何群论中得出调查思路。我们还将讨论一些未解决的问题。