Logic Seminar
Denis Osin
Vanderbilt University
A topological zero-one law and elementary equivalence of finitely generated groups
Abstract: The space of finitely generated marked groups, denoted by $\mathcal G$, is a locally compact Polish space whose elements are groups with fixed finite generating sets; the topology on $\mathcal G$ is induced by local convergence of the corresponding Caley graphs. I will describe a necessary and sufficient condition for a closed subspace $\mathcal S\subseteq \mathcal G$ to satisfy the following zero-one law: for any sentence $\sigma$ 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, the zero-one law holds for certain subspaces associated to hyperbolic groups. This leads to the following (somewhat unexpected) corollary: generic limits of non-cyclic, torsion-free, hyperbolic groups are elementarily equivalent. We will discuss other applications and open problems.
Tuesday November 30, 2021 at 4:15 PM in 636 SEO