Groups represented by incidence geometries

TL;DR

This paper develops a framework using incidence geometries to model both inner and outer automorphisms of groups simultaneously, providing a geometric perspective on automorphism groups.

Contribution

It introduces a novel incidence geometric representation theory that captures inner and outer automorphisms within a unified geometric structure.

Findings

Constructed incidence geometric models for dihedral and symmetric groups.

Represented automorphism groups of Platonic solids and classical groups.

Connected incidence geometries to subgroups of free groups with finite outer automorphism groups.

Abstract

The aim of this paper is to use the framework of incidence geometry to develop a theory that permits to model both the inner and outer automorphisms of a group G simultaneously. More precisely, to any group G, we attempt to associate an incidence system whose group of type-preserving automorphisms is Inn(G), the group of inner automorphisms of G, and whose full group of automorphisms is the group Out(G) of outer automorphisms of G, getting what we call an incidence geometric representation theory for groups. Hence, in this setting, the group Inn(G) preserves the types of the associated incidence structure while the group Out(G) is acting non-trivially on the typeset of it, realizing the outer automorphisms of G as correlations. We give examples of incidence geometric representations for the dihedral groups, the symmetric groups, the automorphism groups of the five platonic solids, families of classical groups defined over fields such as the projective linear groups, and finally subgroups of free groups whose outer automorphism group is the largest finite subgroup of their automorphism group.