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Let $\mathcal G$ denote the space of finitely generated marked groups. Wegive equivalent characterizations of closed subspaces $\mathcal S\subseteq\mathcal G$ satisfying the following zero-one law: for any sentence $\sigma$ inthe 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 showthat the zero-one law holds for certain natural spaces associated to hyperbolicgroups and their generalizations. As an application, we obtain that generictorsion-free lacunary hyperbolic groups are elementarily equivalent; the sameclaim holds for lacunary hyperbolic groups without non-trivial finite normalsubgroups. Our paper has a substantial expository component. We givestreamlined proofs of some known results and survey ideas from topology, logic,and geometric group theory relevant to our work. We also discuss some openproblems.

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