  Given a finitely presented group $Q,$ we produce a short exact sequence $1\to
N \hookrightarrow G \twoheadrightarrow Q \to 1$ such that $G$ is a torsion-free
Gromov hyperbolic group without the unique product property and $N$ is without
the unique product property and has Kazhdan's Property (T). Varying $Q,$ we
show a wide diversity of concrete examples of Gromov hyperbolic groups without
the unique product property. As an application, we obtain Tarski monster groups
without the unique product property.
