## Combinatorics Seminar

When: Sunday, March 15, 10am
Where: Schreiber 309
Speaker: Ehud Friedgut, Weizmann Institute
Title: A sharp threshold for the collapse of the random
triangular group.
## Abstract:

Consider a random group with n generators, and a (random) collection
of relations of size three. We study the evolution of such a group
where the relations are added one by one at random, or (almost)
equivalently, where each possible relation is chosen independently
at random with probability p, and p grows from 0 to 1. We show the
existence of c=c(n), a function bounded between two constants, such that
if p=(1+o(1))cn^{-3/2} the group is asymptotically almost surely trivial,
whereas if p=(1-o(1))cn^{-3/2} then the group is a.a.s. non-trivial.
In fact, if p is just slightly smaller (I'll define "slightly" in the
talk) then the group is a.a.s. infinite, torsion free and hyperbolic.
In this talk we'll mainly study the geometry of the random group, but
won't assume any prior knowledge regarding this notion.
Joint work with Sylwia Antoniuk and Tomasz Luczak.