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 been well studied are cubulation (or more generally, acting cocompactly on a CAT(0) cube complex without global fixed point) and Property (T). In this setting these are mutually exclusive properties, but the threshold densities are not known. In this talk I'll present bounds for threshold densities for both properties, and discuss how these bounds might be improved.
Random Groups Acting on CAT(0) Cube Complexes
MurphyKate Montee, Carleton College
2022 AWM Research Symposium
Women in Groups, Geometry, and Dynamics