A matrix model for random nilpotent groups

Random presentations and random subgroups: a survey

“…In the density model, at any density d > 0, and in any model where the size of the tuple of relators is not bounded, a random tuple h generically presents a perfect group (that is: a group G such that [G, G] = G, or equivalently, a group whose abelian quotient is trivial). [15]: it is well known that every torsion-free nilpotent group embeds in U n (Z) for some n ≥ 2, where U n (Z) is the group of upper-triangular matrices with entries in Z and diagonal elements equal to 1. If 1 ≤ i < n, let a i,n be the matrix in U n (Z) with coefficients 1 on the diagonal and on row i and column i + 1, and all other coefficients 0.…”