Hyperbolicity and cubulability are preserved under elementary equivalence

Abstract

The following properties are preserved under elementary equivalence, among finitely generated groups: being hyperbolic (possibly with torsion), being hyperbolic and cubulable, and being a subgroup of a hyperbolic group. In other words, if a finitely generated group $G$ has the same first-order theory as a group possessing one of the previous properties, then $G$ enjoys that property as well.