Random nilpotent groups I

Matthew Cordes, Moon Duchin, Yen Duong, Meng-Che Ho, Andrew P. Sánchez

Published 2015-06-03Version 1

We study random nilpotent groups in the well-established style of random groups, by choosing relators uniformly among freely reduced words of (nearly) equal length and letting the length tend to infinity. Whereas random groups $\Gamma=F_m/R$ are quotients of a free group by such a random set of relators, random nilpotent groups are formed as corresponding quotients $G=N_{s,m}/R$ of a free nilpotent group. We establish results about the distribution of ranks for random nilpotent groups, and about the distribution of group orders for some finite-order cases. We also study the probabilities of obtaining an abelian or a cyclic group. For example, for balanced presentations (number of relators equal to number of generators), the probability that a random nilpotent group is abelian can be calculated for each rank $m$, and approaches $84.69...\%$ as $m\to\infty$. Further, up to probability, abelian implies cyclic in this setting. We find the precise vanishing threshold for random nilpotent groups---the analog of the famous density one-half theorem for random groups---by studying the abelianization. A random nilpotent group is trivial if and only if the corresponding random group is perfect, i.e., is equal to its commutator subgroup, so this gives a precise threshold at which random groups are perfect. Finally, we describe how to lift results about random nilpotent groups to obtain more general information about the lower central series of standard random groups.