New York Journal of Mathematics
Volume 28 (2022), 1365-1371


Phillip Harris

Random nilpotent groups of maximal step

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Published: October 9, 2022.
Keywords: random groups, nilpotent groups, lower central series.
Subject [2010]: 20D15, 20P05, 60B15.

Let G be a random torsion-free nilpotent group generated by two random words of length L in Un(Z). Letting L grow as a function of n, we analyze the step of G, which is bounded by the step of Un(Z). We prove a conjecture of Delp, Dymarz, and Schaffer-Cohen, that the threshold function for full step is L = n2.


We thank Tullia Dymarz for suggesting this problem and for many helpful discussions. This research was supported by NSF grants DMS-2037851 and DMS-1902173.

Author information

Phillip Harris:
101-05 Van Vleck Hall
University of Wisconsin
Madison, WI 53706, USA