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Cubic maps from the group of order $3$

Vadim Alekseev, Andreas Thom

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 ${\rm PSL}_3(\mathbb Z[ω])$, where $ω$ 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.

Cubic maps from the group of order $3$

Abstract

The purpose of this note is to classify unital cubic maps from the cyclic group of order into an arbitrary non-abelian group. We show that the universal group admitting a unital cubic map from the cyclic group of order is infinite, give a concrete presentation and provide an infinite representation of it in , whose image is an arithmetic lattice commensurable with , where 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 whose image generates the group.
Paper Structure (6 sections, 10 theorems, 30 equations)

This paper contains 6 sections, 10 theorems, 30 equations.

Table of Contents

  1. Quadratic maps and $\mathrm{Pol}_2(C_3)$
  2. Classification of cubic maps from $C_3$
  3. Two concrete representations of ${\rm Pol}_3(C_3)$ with arithmetic image
  4. A representation in characteristic zero
  5. A representation in characteristic $3$
  6. Final discussion

Key Result

Theorem 1

The group ${\rm Pol}_3(C_3)$ is isomorphic to the group where $a = \varphi(\sigma), b=\varphi(\tau)$ for the universal unital cubic map $\varphi \colon C_3 \to {\rm Pol}_3(C_3)$. The elements $a,b$ have infinite order and the group ${\rm Pol}_3(C_3)$ contains a non-abelian free subgroup.

Theorems & Definitions (19)

  • Theorem 1
  • Proposition 1
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • proof : Proof of Theorem \ref{['main']}
  • Corollary 1
  • Theorem 2
  • proof : Outline of the proof
  • ...and 9 more