Finite index subgroups without unique product in graphical small   cancellation groups
  We construct torsion-free hyperbolic groups without unique product whose subgroups up to some given finite index are themselves non-unique product groups. This is achieved by generalising a construction of Comerford to graphical small cancellation presentations, showing that for every subgroup $H$ of a graphical small cancellation group there exists a free group $F$ such that $H*F$ admits a graphical small cancellation presentation. 
Dominik Gruber, Alexandre Martin, and Markus Steenbock
math.GR