Random Nilpotent Groups of Maximal Step

TL;DR

This paper investigates the step of random torsion-free nilpotent groups generated by two random words in upper triangular integer matrices, establishing that the threshold for the group to have maximal step is when the word length scales as the square of the matrix size.

Contribution

It proves a conjecture that the threshold for a random nilpotent group to attain full step is when the generating words have length proportional to the square of the matrix size.

Findings

Threshold for full step is at $ ext{length} \, ext{proportional to} \, n^2$

Confirmed conjecture of Delp, Dymarz, and Schafer-Cohen

Analyzed the step growth as a function of word length and matrix size

Abstract

Let $G$ be a random torsion-free nilpotent group generated by two random words of length $\ell$ in $U_n(\mathbb{Z})$. Letting $\ell$ grow as a function of $n$, we analyze the step of $G$, which is bounded by the step of $U_n(\mathbb{Z})$. We prove a conjecture of Delp, Dymarz, and Schafer-Cohen, that the threshold function for full step is $\ell = n^2$.