Cubulating Random Groups at Densities less than 3/14

MurphyKate Montee (Carleton)

Abstract: Random groups are one way to study "typical" behavior of groups. In the Gromov density model, we often find a threshold density above which a property is satisfied with probability 1, and below which it is satisfied with probability 0. Two properties of random groups that have studied are cubulation and Property (T). In this setting these are mutually exclusive, but the threshold densities are not known. In this talk I'll present a method to demonstrate cubulation on groups with density less than 3/14, and discuss how this might be extended to demonstrate cubulation for densities up to 1/4. In particular, I will describe a construction of walls in the Cayley complex X which give rise to a non-trivial action by isometries on a CAT(0) cube complex. This extends results of Ollivier-Wise and Mackay-Przytycki at densities less than 1/5 and 5/24, respectively.