A Matrix Model for Random Nilpotent Groups
Kelly Delp, Tullia Dymarz, Anschel Schaffer-Cohen · International Mathematics Research Notices · 2017
We study random torsion-free nilpotent groups generated by a pair of random words of length |$\ell$| in the standard generating set of |$U_n(\mathbb{Z})$|. Specifically, we give asymptotic results about the step properties of the group when the lengths of the generating words are functions of |$n$|. We show that the threshold function for asymptotic abelianness is |$\ell = c \sqrt{n}$|, for which the probability approaches |${\rm e}^{-2c^2}$|, and also that the threshold function for having full step, the same step as |$U_n(\mathbb{Z})$|, is between |$c n^2$| and |$c n^3$|. We conjecture that the actual threshold function for full step is |$n^2$|. Finally we discuss an approach to give bounds for intermediate steps between step one (abelian) and full step.