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