Random Nilpotent Groups I

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 T = Fm/<<R>> are quotients of a free group by such a random set of relators, random nilpotent groups are formed as corresponding quotients G = Ns,m/ <<R>> of a free nilpotent group. Using arithmetic uniformity for the random walk on Zm and group-theoretic results relating a nilpotent group to its abelianization, we are able to deduce statements about the distribution of ranks for random nilpotent groups from the literature on random lattices and random matrices. We obtain results about the distribution of group orders for some finite-order cases as well as the probability that random nilpotent groups are abelian. 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 --> ∞ ⁠. Further, abelian implies cyclic in this setting (asymptotically almost surely). Considering the abelianization also yields the precise vanishing threshold for random nilpotent groups—the analog of the famous density one-half theorem for random groups. A random nilpotent group is trivial if and only if the corresponding random group is perfect, that is, is equal to its commutator subgroup, so this gives a precise threshold at which random groups are perfect. More generally, we describe how to lift results about random nilpotent groups to obtain information about the lower central series of standard random groups.