Hyperbolicity and Cubulability Are Preserved Under Elementary Equivalence

TL;DR

This paper proves that certain geometric properties of finitely generated groups, such as hyperbolicity and cubulability, are preserved under elementary equivalence, meaning they are first-order definable and invariant.

Contribution

It establishes that hyperbolicity, cubulability, and subgroup relations within hyperbolic groups are first-order properties preserved under elementary equivalence.

Findings

Hyperbolicity is preserved under elementary equivalence.

Cubulability is preserved under elementary equivalence.

Subgroup relations within hyperbolic groups are preserved.

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 property, then G enjoys this property as well.