Finite covers of a product of surfaces with bounded rank and arbitrarily large systole

We construct finite covers of a fixed product of two closed hyperbolic surfaces of genus two whose fundamental groups are generated by at most fifteen elements and whose injectivity radii tend to infinity. The construction uses fibre products over finite groups. These covers are closed aspherical four-manifolds with universal cover ${\mathbb{H}}^2\times {\mathbb{H}}^2$. They give counterexamples to a conjecture of Avramidi and Delzant for symmetric spaces of higher rank, in the case ${\mathbb{H}}^2\times {\mathbb{H}}^2$.