Random nilpotent groups I

TL;DR

This paper investigates the properties of random nilpotent groups formed as quotients of free nilpotent groups by randomly chosen relators, revealing new phenomena distinct from standard random groups due to their nilpotent structure.

Contribution

It introduces a novel model for studying random nilpotent groups and explores phenomena not observable in traditional random group models.

Findings

Identification of phase transitions specific to nilpotent groups

Discovery of new structural properties in random nilpotent groups

Contrast with behavior of standard random groups

Abstract

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 are quotients of a free group by such a random set of relators, random nilpotent groups are formed as corresponding quotients of a free nilpotent group. This model reveals new phenomena because nilpotent groups are not "visible" in the standard model of random groups (due to the sharp phase transition from infinite hyperbolic to trivial groups).