A topological zero-one law and elementary equivalence of finitely generated groups

Let $\mathcal{G}$ denote the space of finitely generated marked groups. We give equivalent characterizations of closed subspaces $\mathcal S\subseteq \mathcal G$satisfyingthefollowingzero-onelaw:foranysentence$\sigma$ in the infinitary logic ${\mathcal{L}}_{{\omega }_1,\omega }$, the set of all models of $\sigma$ 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.