A sharp threshold for collapse of the random triangular group
  The random triangular group $\Gamma(n,p)$ is the group given by a random group presentation with $n$ generators in which every relator of length three is present independently with probability $p$. We show that in the evolution of $\Gamma(n,p)$ the property of collapsing to the trivial group admits a very sharp threshold. 
Sylwia Antoniuk, Ehud Friedgut, and Tomasz {\L}uczak
math.GR math.CO