Finite index subgroups without unique product in graphical small cancellation groups

fasya kamila muftia;

For every integer k  1, we construct a torsion-free hyperbolic group without unique product all of whose subgroups up to index k are themselves non-unique product groups. This is achieved by generalizing 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