Incidence geometries with trialities coming from maps with Wilson trialities

TL;DR

This paper connects Wilson triality in reflexible maps with classical triality in geometry, constructing incidence geometries with trialities from maps and exploring their group-theoretic properties.

Contribution

It establishes a method to derive incidence geometries with trialities from maps exhibiting Wilson triality and extends existing results on the groups associated with such maps.

Findings

Constructed incidence geometries with triality from Wilson triality maps

Proved existence of maps with Wilson triality but no duality for groups ${ m L}_2(q^3)$

Extended results relating Wilson triality to classical triality in geometry

Abstract

Triality is a classical notion in geometry that arose in the context of the Lie groups of type $D_4$. Another notion of triality, Wilson triality, appears in the context of reflexible maps. We build a bridge between these two notions, showing how to construct an incidence geometry with a triality from a map that admits a Wilson triality. We also extend a result by Jones and Poulton, showing that for every prime power $q$, the group ${\rm L}_2(q^3)$ has maps that admit Wilson trialities but no dualities.