Rips construction without unique product
  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. 
Goulnara Arzhantseva and Markus Steenbock
math.GR