Cubic maps from the group of order $3$

TL;DR

This paper classifies unital cubic maps from the cyclic group of order 3 into non-abelian groups, revealing an infinite universal group with a concrete representation linked to arithmetic lattices.

Contribution

It introduces the universal group for unital cubic maps from C3, provides its presentation, and connects it to arithmetic lattices in PSL_3(C).

Findings

Universal group is infinite and admits a concrete representation.

Existence of finite nilpotent groups with unital cubic maps from C3.

The image of the universal map is an arithmetic lattice in PSL_3(C).

Abstract

The purpose of this note is to classify unital cubic maps from the cyclic group of order $3$ into an arbitrary non-abelian group. We show that the universal group admitting a unital cubic map from the cyclic group of order $3$ is infinite, give a concrete presentation and provide an infinite representation of it in ${\rm PSL}_3(\mathbb C)$, whose image is an arithmetic lattice commensurable with ${\rm PSL}_3(\mathbb Z[\omega])$, where $\omega$ is a primitive cube root of unity. As a consequence we obtain the existence of finite nilpotent groups of arbitrarily large nilpotency class admitting a unital cubic map from $C_3$ whose image generates the group.