On Graphs, Groups and Geometry

TL;DR

This paper explores the concept of natural metric spaces and groups, demonstrating how various algebraic structures can be characterized by unique isometric group operations and examining their properties and extensions.

Contribution

It introduces the notion of natural groups and graphs, showing how many common groups and graph products are natural, and discusses methods to extend non-natural groups to become natural.

Findings

Graphical regular representations are always natural.

Certain graph products preserve naturalness.

Some groups can be extended to become natural groups.

Abstract

A metric space (X,d) is declared to be natural if (X,d) determines an up to isomorphism unique group structure (X,+) on the set X such that all the group translations and group inversion are isometries. A group is called natural if it emerges like this from a natural metric. A simple graph X is declared to be natural if (X,d) with geodesic metric d is natural. We look here at some examples and some general statements like that the graphical regular representations of a finite group is always a natural graphs or that the direct product on groups or the Shannon product of finite graphs preserves the property of being natural. The semi-direct product of finite natural groups is natural too as they are represented by Zig-Zag products of suitable Cayley graphs. It follows that wreath products preserve natural groups. The Rubik cube for example is natural. Also free products of finitely generated natural groups are natural. A major theme is that non-natural groups often can be upgraded to become natural by extending them to become Coxeter groups. Examples of non-natural groups are cyclic groups whose order is divisible by 4, the quaternion group, the integers, the lamplighter group, the free groups or the group of p-adic integers. The prototype feature is to extend the integers and get the infinite dihedral group, replacing the single generator by two free reflections. We conclude with a short discussion of the hypothesis of using the dihedral group as a physical time in dynamical system theory.