Free and Hyperbolic Groups are not Equational

TL;DR

The paper demonstrates that in free and torsion-free hyperbolic groups, there exist definable sets outside the Boolean algebra of equational sets, showing these theories are not equational.

Contribution

It provides the first example of definable sets in these groups that are not in the Boolean algebra of equational sets, challenging previous assumptions.

Findings

Existence of definable sets outside the Boolean algebra of equational sets

Theories of free and torsion-free hyperbolic groups are not equational

Counterexamples to the equational nature of these theories

Abstract

We give an example of a definable set in every free or torsion-free (non-elementary) hyperbolic group that is not in the Boolean algebra of equational sets. Hence, the theories of free and torsion-free (non-elementary) hyperbolic groups are not equational in the sense of G. Srour.