A matrix model for random nilpotent groups

TL;DR

This paper investigates the properties of random torsion-free nilpotent groups generated by two random words in the upper triangular integer matrices, identifying thresholds for abelianness and maximal step as functions of matrix size.

Contribution

It provides asymptotic thresholds for when such groups are almost surely abelian or have full step, advancing understanding of random nilpotent group structures.

Findings

Threshold for asymptotic abelianness is at length c√n with probability e^{-2c^2}

Full-step property occurs between lengths c n^2 and c n^3

Results connect group properties with the length of generating words and matrix size

Abstract

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 $e^{-2c^2}$, and also that the threshold function for having full-step, the same step as $U_n(\mathbb{Z})$, is between $c n^2$ and $c n^3$.