Report on Freely Representable Groups

TL;DR

This report provides a comprehensive, self-contained overview of freely representable groups, highlighting their mathematical properties, applications in geometry and number theory, and introducing new perspectives and results.

Contribution

It synthesizes existing knowledge on freely representable groups, introduces new viewpoints, terminology, and proofs, and offers a detailed, accessible development of the subject.

Findings

Freely representable groups are characterized by their linear representations with only the zero vector fixed.

These groups are used in classifying spaces of constant positive curvature.

Recent applications include their role in algebraic number theory and norm relations.

Abstract

This report is an account of freely representable groups, which are finite groups admitting linear representations whose only fixed point for a nonidentity element is the zero vector. The standard reference for such groups is Wolf (1967) where such groups are used to classify spaces of constant positive curvature. Such groups also arise in the theory of norm relations in algebraic number theory, as demonstrated recently by Biasse, Fieker, Hofmann, and Page (2020). This report aims to synthesize the information and results from these and other sources to give a continuous, self-contained development of the subject. I introduce new points of view, terminology, results, and proofs in an effort to give a coherent, detailed, self-contained, and accessible narrative.