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We consider two random group models: the hexagonal model and the squaremodel, defined as the quotient of a free group by a random set of reduced wordsof length four and six respectively. Our first main result is that in thismodel there exists a sharp density threshold for Kazhdan's Property (T) and itequals 1/3. Our second main result is that for densities < 3/8 a random groupin the square model with overwhelming probability does not have Property (T).Moreover, we provide a new version of the Isoperimetric Inequality thatconcerns non-planar diagrams and we introduce new geometrical tools toinvestigate random groups: trees of loops, diagrams collared by a tree of loopsand specific codimension one structures in the Cayley complex, called benthypergraphs.

Bent walls for random groups in the square and hexagonal model