Bent walls for random groups in the square and hexagonal model

Abstract:

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 the hexagonal model there exists sharp density threshold for Kazhdan’s Property (T) and it equals $\frac {1}{3}$. Our second main result is that for densities $< \frac {3}{8}$ 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.