Cubulating random groups in the square model

Our main result is that for densities < 3/10 a random group in the square model has the Haagerup property and is residually finite. Moreover, we generalize the Isoperimetric Inequality to some class of non-planar diagrams and, using this, we introduce a system of modified hypergraphs providing the structure of a space with walls on the Cayley complex of a random group. Then we show that the natural action of a random group on this space with walls is proper, which gives the proper action of a random group on a CAT(0) cube complex.