Event
Tomasz, Odrzygozdz
Wednesday, November 2, 2016 15:00to16:00
Burnside Hall
Room 920, 805 rue Sherbrooke Ouest, Montreal, QC, H3A 0B9, CA
Sharp threshold for Property (T) in the hexagonal model for random groups.
A random group in the hexagonal model is a group given by the presentation where R is a set of random words of length 6 over the set S. We consider properties of such a group as the cardinality of S goes to infinty. Our main goal in the presentation is to prove that as the cardinality of the set R increases (we increase the density defined as d =(1) *log_{|S|}(|R|)) there is a sharp transition between not having Property (T) and having Property (T) for such group (this threshold is at density d=1/3). First we will present a quick survey about what is known at the moment about Property (T) for random groups. To proof thie main result we will present a new method of constructing some good system of walls on the Cayley complex of a random group. This will allows us to find a proper action of a random group on a CAT(0) cube complex. The main idea behind constructing our system of walls is to take hypergraphs in the Cayley complex (as Wise and OIllivier did in their paper about cubulating random groups) and then correct them to make them embedded trees.